Difference of linkage between spot and futures
markets in Nikkei225 stock index and DJIA index
Takanori Usuia
Isao Shojib
a
Graduate School of System and Information Engineering, University of Tsukuba,
Tenoudai 1-1-1, Tsukuba science city, Ibaraki: 305-8573, Japan.
Mail address: hiromachi492@yahoo.co.jp
b
Graduate School of System and Information Engineering, University of Tsukuba,
Tenoudai 1-1-1, Tsukuba science city, Ibaraki: 305-8573, Japan.
Mail address: shoji@sk.tsukuba.ac.jp
Phone number: 029-853-5094
Fax number: 029-853-5094
Abstract
This paper investigated the difference of linkage between spot and futures in
Nikkei225 stock index and Dow Jones Industrial Average (DJIA) index. Especially we
focused on information transmission with volatility spillover effect. We used bivariate
generalized autoregressive conditional heteroskedasticity (GARCH) model and
bivariate exponential GARCH (EGARCH) model to estimate. Daily data from
2001/01/03 to 2006/03/30 were chosen. From empirical result, it became clear that
Nikkei225 stock index followed bivariate GARCH, while DJIA did bivariate EGARCH.
Additionally in Nikkei225 stock index, futures market was more dominantly to have an
affect on spot market and reverse case doesn’t exist. Contrary to Nikkei225 stock index,
volatility spillover from spot to futures in DJIA index is stronger than reverse case. But
effect from futures to spot couldn’t be ignored.
Introduction
The main purpose of this paper is to investigate the differences of markets structure
between linkage of Nikkei225 index spot on its futures and that of Dow Jones Industrial
Average (DJIA) index spot and futures with bivariate GARCH and EGARCH. Nikkei225
and DJIA are very famous stock indices for judging the conditions of Japanese and U.S.
economics. Consequently enough information flows in their spot and futures. To examine
information transmission, we adopted Volatility spillover effect in this time. Volatility
spillovers are important because volatility is also a source of information. French & Roll
(1986); and Ross, (1989) show that variance is an important source of information.
French and Roll found that asset prices are much more volatile during exchange trading
hours than at other times and that this divergence is caused by differences in the flow of
information. Ross proved that asset price volatility is related to the rate of information
flow in competitive asset markets. In a study of volatility spillovers among similar assets,
there are many papers which considered information transmission between stock index
and index futures markets. Tse (1999) used Dow Jones Industrial Average (DJIA) with
bivariate EGARCH model. This study is reported that spot and futures have strong
effect to the other. Additionally the effect from futures market to spot is stronger than
reverse case. Iihara(1996) examined Nikkei225 index spot and its futures with bivariate
GARCH model. This study reports that after March 1991, there is no volatility spillover
effect between spot and futures. Bhar (2001) used Australian stock index spot (AOI) and
futures (SPI) with bivariate EGARCH-X model. The remark in this study is that AOI
and SPI are highly correlated and information transmission from futures to spot and
reverse case exist. Meneu and Torro (2003) examined Spanish stock index (IBEX-35) and
its futures with several Asymmetric Covariance model. In the conclusion, it mentions
that main source of uncertainty comes from the spot market, and it is transmitted to the
futures market. Reverse volatility spillover also exists, but is far less important. This
conclusion is completely different from other result. So and Tse (2004) examined 3 assets
relationships; Hang Seng Index spot which is the stock index in Hong Kong, its futures
and tracker fund of that Index. The result shows that there is strong bi-directional
volatility spillover between spot and futures, but tracker fund isn’t. Zhong, Darrat and
Otero (2004) examined Mexican stock index spot and futures with developed BEKK
model. It gets the conclusion that Mexican stock index futures market plays a part of
price discovery function of its spot market that is to say futures market affect spot
market dominantly.
From the result of previous study, the significance of this investigation is to compare
the differences of linkages between Nikkei225 stock index spot-futures markets and
DJIA spot-futures markets, because the linkage between spot and futures is different in
each country, and there are few studies to compare the linkage of spot and futures in
Japan and U.S. with same model and frequency. Our result shows that bivariate GARCH
is suitable for that of Nikkei225 spot-futures markets but bivariate EGARCH is suitable
for describing to linkage of DJIA spot-futures markets. From the best estimate result, it
is true that volatility spillover from Nikkei225 stock index futures to its spot exists
clearly, but the reverse flow doesn’t. While in DJIA index, its spot spillovers volatility to
futures stronger, but reverse flow also exist and it can’t be ignored.
Section 2 presents data and sumary statistics. Section 3 presents empirical
methodology. The empirical result is presented in section 4. And section 5 concludes the
study.
Data and Summary statistics
TABLEⅠ
Statistics of Daily log return rate of Nikkei225 stock index spot and futures
Summary statistic
Spot
Futures
Mean(10 )
0.112
0.114
Standard deviation(10-1)
0.148
0.135
Skewness
-0.0631
-0.264
Kurtosis
1.460
2.942
Maximum
0.072
0.080
Minimum
-0.069
-0.102
-3
correlation
0.958
Statistics of Daily log return rate of DJIA index spot and futures
Summary statistic
Spot
Futures
Mean(10 )
0.011
0.011
Standard deviation(10-1)
0.111
0.112
Skewness
-0.011
0.169
Kurtosis
4.511
4.541
Maximum
0.064
0.062
Minimum
-0.077
-0.076
-3
correlation
0.955
Note: The term is from 2001/01/03 to 2006/03/30 those of holidays in the term are omitted. The total number of
observation in the sample is 1290 in Nikkei225 and 1313 in DJIA.
In this study, we use the daily data of closing price of Nikkei225 stock index spot and its
futures and also DJIA index spot and futures. The term is from 2001/01/03 to 2006/03/30
those of holidays in the term are omitted. The total number of observation in the sample
is 1290 in Nikkei225 and 1313 in DJIA. We write out the summary statistics of the spot
and futures in Nikkei225 stock index and DJIA index in TableⅠ. From TableⅠ, both
Nikkei225 and DJIA are highly correlated and Nikkei225 stock index futures gets higher
Skewness, and Kurtosis compare to underlying asset, while in DJIA, its futures gets
higher skewness than spot.
Empirical Methodology
First introduced by Engle (1982) and Bollerslev (1986), the ARCH and GARCH models
are often used to describe time-variation in the volatility of financial returns. In this
time we posit the following bivariate GARCH(1,1) model and bivariate EGARCH(1,1)
model for the joint process governing the stock index and futures as presenting
conditional return and volatility:
Bivariate GARCH
R  μ  εt
(1)
εst ,t 
Φt 1～N (0, H t )
εt  
ε
ft
,
t


 hst
H t  
 h fs
hsf 

h ft 
(2)
hst ,t  ωst αstε2st ,t 1 βst hst ,t 1 γstε2ft ,t 1
(3)
h ft ,t  ωft αftε2ft ,t 1 βft h ft ,t 1 γftε2st ,t 1
(4)
hsf ,t  h fs ,t ρ(hst ,t 1h ft ,t 1 )1 / 2
(5)
where st refers to spot and ft refers to futures. the log return rate vector for the spot and
futures series is given by Rt=[Rst,t Rft,t], the residual vectors by εt=[εst,t εft,t], and the
conditional covariance matrix by Ht, where {Ht}ij=hij,t, for i,j=st, ft. the parameter vectors
are defined as μ=[μ st μ ft],. Φt-1 is the set of information available at time t－1.
Equation (1) models the return of spot and futures as an AR(0) process. The number of
parameter was estimated by Box-Jenkins method. Conditional on the dependence in
means the spot and futures return series are assumed to have a bivariate normal
distribution covariance matrix Ht. Equation (2) to (5) model the diagonal elements of the
conditional covariance matrix Ht as a function of the diagonal elements of the conditional
covariance matrix of the past period. In these equations we specify the influence from
each market to adopt the coefficients of εst in hft and vice versa. The benefits to adopt
them are to be able to analyze from influences from each market clearly. In Equation (3)
and (4), αst, αft, denote the market-specific volatility clustering, and βst, βft denote
volatility persistence. Volatility spillover effect is expressed in γst, γft. For example,
when γst is larger thanαst, it means that the innovation from futures market is more
dominantly to form spot price than that in spot markets. In equation (5) ρ means
correlation of two series. In this time we use constant correlation model which is
suggested by Bollerslev(1990) at first. But it doesn’t guarantee that correlation is
constant. Baur (2004), Darbar and Deb(2002) and Lien and Yang (2006) suggest the
correlation varying model.
Bivariate EGARCH
R  μ  εt
(6)
εst ,t 
Φt 1～N (0, H t )
εt  
ε
 ft ,t 
 hst
H t  
 h fs
hsf 

h ft 
(7)
ln hst ,t  ωst αst ust ,t 1 θst Gst ,t 1 βst ln hst ,t 1 γst u ft ,t 1 ηst G ft ,t 1
(8)
ln h ft ,t  ωft αft u ft ,t 1 θft G ft ,t 1 βft ln h ft ,t 1 γft u st ,t 1 ηft Gst ,t 1
(9)
Git  uit  E uit , uit εit
(10)
hsf ,t  h fs ,t ρ(hst ,t 1h ft ,t 1 )1 / 2
hit , i  st , ft
(11)
As well as bivariate GARCH model, where the log return rate vector for the spot and
futures series is given by Rt=[Rst,t Rft,t], the residual vectors by εt=[εst,t εft,t], and the
conditional covariance matrix by Ht, where {Ht}ij=hij,t, for i,j=st, ft. the parameter vectors
are defined as μ=[μ st μ ft]. Φt-1 is the set of information available at time t－1.
Equation (6) models the return of spot and futures as an AR(0) process. The number of
parameter was estimated by Box-Jenkins method. Conditional on the dependence in
means the spot and futures return series are assumed to have a bivariate normal
distribution covariance matrix Ht. Equation (7) to (9) model the diagonal elements of the
conditional covariance matrix Ht as a function of the diagonal elements of the
conditional covariance matrix of the past period. In these equations we specify the
influence from each market to adopt the coefficients of Gst and ust in hft and vice versa.
In Equation (8) and (9), αst, αft, denote the market-specific volatility clustering, and
βst, βft denote volatility persistence. Volatility spillover effect is expressed in γst, γft.
θst, θft andηst, ηft express the effect of asymmetric movement of volatility from own
market and the other market respectively. In equation (11) ρ means correlation of two
series, which is constant.
Given a sample of T daily returns, the parameter of the multivariate system (1) to (5)
and (6) to (11) are estimated by computing the conditional log likelihood function for each
time period as
1
1
Lt (θ)   log 2π log | H t (θ) |  εt (θ)H t1 (θ)εt (θ)
2
2
T
L(θ)   Lt (θ)
(12)
(13)
t 1
where θ is the vector of all parameters. We implements the algorithm which developed
by Berndt, Hall, Hall and Hausman (BHHH algorithm) (1974) where normally
distributed returns are assumed.
Empirical result
At first, we estimated both markets linkage between spot and futures with bivariate
GARCH and bivariate EGARCH. The result is shown in TableⅡ. From this table, it
implicates that linkage between spot and futures in Nikkei225 is suitable for fitting
bivariate GARCH, while that in DJIA is suitable for fitting bivariate EGARCH. This
difference implies that Nikkei225 spot and futures markets don’t contain asymmetric
effect in the markets whereas DJIA ones do.
TABLEⅡ
Comparison with bivariate GARCH and EGARCH in Nikkei225 stock index and DJIA index
Nikkei 225 stock index
Log-likelihood
AIC
DJIA index
BIC
Log-likelihood
AIC
BIC
bivariate GARCH
-6322
6340 6350
-5964
5982 5982
bivariate EGARCH
-6391
6417 6410
-5801
5827 5823
Note: Bivariate GARCH
Bivariate EGARCH
εt | Φt 1～N (0, Ht )
 hst hsf 

H t  
 h fs h ft 
εt | Φt 1～N (0, Ht )
 hst hsf 

H t  
 h fs h ft 
hst ,t  ωst αstε2st ,t 1 βst hst ,t 1 γstε2ft ,t 1
ln hst ,t  ωst αst ust ,t 1 θst Gst ,t 1 βst ln hst ,t 1 γst u ft ,t 1 ηst G ft ,t 1
h ft ,t  ωft αftε2ft ,t 1 βft h ft ,t 1 γftε2st ,t 1
ln h ft ,t  ωft αft u ft ,t 1 θft G ft ,t 1 βft ln h ft ,t 1 γft u st ,t 1 ηft Gst ,t 1
hsf , t  h fs , t ρ(hst , t 1h ft , t 1 )1 / 2
Git  uit  E uit , uit εit hit , i  st , ft
hsf , t  h fs , t ρ(hst , t 1h ft , t 1 )1 / 2
TableⅢ and TableⅣ are the estimation result with the best model in Nikkei225 and
DJIA respectively. From TableⅢ, αft is lager than αst. It means that innovation in
futures market is bigger than spot market. Moreover, βft is also lager, so volatility
persistence is stronger in futures market. Correlation ρ is very high, and these are
highly correlated. The vital point in this table is γ. Because γft which means volatility
spillover effect from spot to futures is insignificant contrary to γst. This implies that
volatility spillover only exists from futures to spot but reverse case doesn’t. Moreover,
compare with α st and γ st, γ st is larger than α st, so futures market is more
dominantly to form spot market in Nikkei225 stock index.
TABLEⅢ
Estimation result of Nikkei225 with bivariate GARCH
Spot
Futures
ω
10.44*(5.805)
7.593*(4.496)
α
0.06**(2.423)
0.116*(4.719)
β
0.815*(44.803)
0.844*(53.658)
γ
0.078*(3.147)
0.013(0.551)
0.961*(447.575)
ρ
Note: Bivariate GARCH
εt | Φt 1～N (0, Ht )
 hst hsf 

H t  
 h fs h ft 
hst ,t  ωst αstε2st ,t 1 βst hst ,t 1 γstε2ft ,t 1
h ft ,t  ωft αftε2ft ,t 1 βft h ft ,t 1 γftε2st ,t 1
hsf , t  h fs , t ρ(hst , t 1h ft , t 1 )1 / 2
γst and γft indicate the volatility spillover effect from the other market and ρ indicates the correlation between
spot and futures market. An *(**) indicate statistical significance at the 1% (5%) level and t- statistics are indicated
in parentheses.
TableⅣ is estimation result of DJIA index. This result shows that αst is larger thanα
ft,
and βst which means volatility persistence is also stronger in spot market. In terms
of the volatility spillover effect from the other market which is expressed by γ, the
magnitude of γft is larger than γst. This implicates that volatility generated in spot
market spillovers to futures market strongly. Additionally negative innovation in spot
market makes futures market volatility increase. Compare with the magnitude of α
and γ, αst is larger than γst andγft is larger than αft. From the result, spot market
is more dominantly to form futures market in DJIA. θ implicates asymmetric effect
and both θst and θft are significant, so there is asymmetric effect which affect from
own market. But η isn’t significant though it express asymmetric effect affected from
the other market. Consequently, volatility asymmetric is caused by own market and the
other market doesn’t affect anything. Furthermore, asymmetric effect is stronger in
spot market. Correlation ρis as high as that in Nikkei, so spot and futures are also
highly correlated in DJIA.
TABLEⅣ
Estimation result of DJIA with bivariate EGARCH
Spot
Futures
ω
0.465*(7.218)
0.465*(6.485)
α
-0.439*(-15.603)
0.181*(3.254)
β
0.901*(56.967)
0.898*(59.021)
γ
0.027*(7.364)
-0.363*(-6.24)
θ
0.288*(6.971)
0.196*(4.183)
η
-0.002(-0.663)
0.028(0.489)
*
ρ
0.945 (303.723)
Note: Bivariate EGARCH
εt | Φt 1～N (0, Ht )
 hst hsf 

H t  
 h fs h ft 
ln hst ,t  ωst αst ust ,t 1 θst Gst ,t 1 βst ln hst ,t 1 γst u ft ,t 1 ηst G ft ,t 1
ln h ft ,t  ωft αft u ft ,t 1 θft G ft ,t 1 βft ln h ft ,t 1 γft u st ,t 1 ηft Gst ,t 1
Git  uit  E uit , uit εit hit , i  st , ft
hsf , t  h fs , t ρ(hst , t 1h ft , t 1 )1 / 2
γst and γft indicate the volatility spillover effect from the other market, θst and θft indicate the volatility
asymmetric effect from own market, ηst and ηft express the volatility asymmetric effect from the other market,
and ρ indicates the correlation between spot and futures market. An *(**) indicate statistical significance at the
1% (5%) level and t- statistics are indicated in parentheses.
FigureⅠ and FigureⅡ are the graph which plot daily volatility of Nikkeo225 stock
index and DJIA index respectively. Horizontal axis denotes date and vertical axis
denotes volatility. From figureⅠ, the mean of volatility of spot in Nikkei225 stock index
is 213.107 and that of futures is 223.755. The mean of futures volatility is larger than
that of spot. It means that futures market is more volatile than spot market and its
volatility would spillover to spot market.
1800
1600
1400
volatility
1200
spot
futures
1000
800
600
400
200
0
200
0/1
0/1
200
2/2
/13
200
3/6
/28
200
4/1
1/9
200
6/3
/24
date
Figure 1
daily volatility of NIkkei225
On the other hand, in FigureⅡ, the mean of volatility of spot is 118.154 in DJIA index
and that of futures is112.407. The mean of spot volatility is larger than that of futures
but futures daily volatility sometimes excesses spot. Particularly, when both volatilities
are high, futures volatility tends to become larger than spot. From the Figures of
Nikkei225 stock index and DJIA index, they got completely different result.
1800
1600
1400
volatility
1200
spot
1000
futures
800
600
400
200
0
200
0/
10/
1
200
2/2
/13
200
3
200
/6/
28
4/1
1/9
200
6/3
/24
date
Figure 2
daily volatility of DJIA
Conclusion
This study examined the differences of markets structure between linkage of
Nikkei225 stock index spot on its futures and that of Dow Jones Industrial Average
(DJIA) index with using Daily data from2001/01/03 to 2006/03/30. In measuring
volatility spillover between spot and futures, this study employed the bivariate GARCH
and bivariate EGARCH methodology. Particularly we focused on which model is suitable
for each country’s markets and which indices (spot and futures) get dominant effect and
how much it is. The derived empirical results show that Nikkei225 stock index is
suitable for fitting bivariate GARCH, and futures market dominantly form spot market
and effect from spot to futures doesn’t exist. On the other hand, DJIA index is suitable
for fitting bivariate EGARCH, and spot market is stronger than futures market to
spillover volatility to the other markets but the reverse can’t be ignored.
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