2015 International Conference on Management Science & Engineering (22th)
October 19-22, 2015
Dubai, United Arab Emirates
Dynamics and Information Transmission between Stock Index
and Stock Index Futures in China
WANG Chao-you，KOU Yi
School of Management, Harbin Institute of Technology, Harbin 150001, P.R. China
Abstract: This research employs the multivariate
GARCH models to investigate the volatility transmission
between the stock market and index futures market in
China. The daily returns of the latest four years are tested
as sample data. Three different multivariate GARCH
models (BEKK, CCC and DVEC) are estimated and
compared. The BEKK model works best and is able to
capture the bidirectional transmission between the two
markets. The conditional correlations are close to 1 for
most of the time, which shows the strong linkage of the
two prices. In addition, we find the variance of the
futures is larger than the spot market. Above results
indicate the futures perform better compared to its earlier
stage. The information transmission between the two
markets proves that the futures’ ability in price
discovery.
Keywords: multivariate GARCH, volatility
transmission, stock index futures
1 Introduction
China launched its first stock index futures in 2010.
This was the first time that investors could sell short in
Chinese market. It also helps the price discovery and
provides the tool to hedge risk. Five years later, the index
futures is now the mostly traded futures in China As the
two markets are highly connected, the information and
risk transmission transfer bidirectionally and quickly.
The stock futures is designed to improve the price
discovery and lower the risk of the spot market. The
performance of the futures is of great significance.
Therefore, the relationship between the index futures and
its underlying spot market attracts many researchers.
Abundant researches have been done to investigate
the relationship between the index futures and the stock
index across the world. The results of different markets
differ. Garbade and Silber (1983)[1] test the prediction
ability of the basis on the price of spot and futures
markets in the U.S. market. They find that the futures
dominates cash markets for most commodities. However,
the integration of two markets over short times is much
weaker in grains than gold and silver. Cakici and
Chatterjee (1991)[2] find the S&P500 index futures
decreases the volatility of the cash market, which proves
the benefit of the futures’ introduction. The result is also
978-1-4673-6513-0/15/$31.00 ©2015 IEEE
found in the interest rate market. Merton (1995)[3]
indicates the futures weaken the asymmetry of the
underlying stock market. In addition, the improvements
in information technology leads to the structure change
in the long run relation. Due to the late introduction of
the Chinese index futures, the researches about the
Chinese market are relatively deficient. Yang, Yang and
Zhou (2012)[4] uses high-frequency data to investigate
the relationship. They find the futures is not mature
enough in its first year due to the high entry barrier. The
cash market still contributes more in the price discovery
process, which is different from its foreign counterparts.
Yan (2011)[5] find the two markets are not causal related
and the futures market does not contribute the price
discovery significantly. Wang and Xie (2013)[6] use high
frequency data to study the cross-correlations. They find
the correlations are strongly multifractal, which also
indicates the immaturity of the futures market. These
studies provide the discovery about the infancy stage of
the Chinese index futures. However, after five years
operation, the trading volume grows rapidly and is
already the highest of all futures in futures market
nationwide. The transaction fee and the entry barrier are
much lower now. The effect of the futures should be
reinvestigated to see the difference from the earlier years
which would be helpful for the development of the
markets.
The multivariate GARCH model is wildly used to
testify the dynamic relationship between multiple
financial assets and in different countries [7-11]. For
example, stock and index futures markets in
Hongkong[12], Korea[13] and UK[14] have been studied. It
is also used to compare the price discovery between
markets[15-17] and jump dynamics[18,19]. It captures the
volatility movements of related assets as well as their
correlations and time-varying information transmission.
This paper attempts to use three varieties of multivariate
GARCH models to test the past four years’ data.
Therefore, the features of different phases can be
explored.
2 Model specification
The bivariate GARCH models are developed from
the univariate GARCH model by Bollerslev (1986)[19]
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and it take the interaction of variances into account. The
ARCH and GARCH models are the mostly used models
in describing the time-varying variance of time series. In
this paper, three multivariate GARCH models (BEKK,
constant conditional correlation, Diagonal VECH) are
used to model the volatility transmission between the
CSI 300 index futures and its underlying spot markets.
All the three models contain a vector autoregression
(VAR) process. The VAR process resembles the ARMA
process in univariate GARCH. The BEEK-GARCH
proposed by Engle and Kroner (1995)[20] can be specified
by the following equations:
RμR
(1)
t =+ a t-1 + t
m
s
Ht =
AA′ + ∑ A i a t-i a′t-i A′i + ∑ B jH t-jB′j (2)
=i 1 =j 1
The VAR process is depicted by Equation (1). Rt is
the return series vector of two assets. The at is a 2×1
vector of residuals. Ht is a 2×2 matrix of the variance
and covariance of the residuals. The A is a symmetric
parameter matrix. Ai and Bj and coefficient matrices,
which capture the moving average and autoregressive
effect of the volatility respectively. The off-diagonal
elements of matrices Ai and Bj provide the information
transmission across the two assets. The BEKK-GARCH
model can capture features of “heavy tails” and
“volatility clustering” like univariate GARCH model. It
also can guarantee that the matrix are positive definite
under very weak conditions.
The equations for the CCC-GARCH and Diagonal
VEC GARCH are similar to the BEKK model. The
DVEC is a straightforward generalization of the
univariate GARCH model to the multivariate case. Every
conditional variance and covariance is a function of all
lagged conditional variances and covariances, as well as
lagged squared returns and cross–products of returns.
m
s
Ht =
A 0 + ∑ A i ⊗ (a t-i a′t-i ) + ∑ B j ⊗ H t-j (3)
=i 1 =j 1
The Equation (3) depicts the DVEC GARCH model
and ⊗ denotes the Hadamard product, that is,
element-by-element multiplication.
The DVEC model can also be described in Equation
(4)-(6) in detail. Equation (5) captures the information
transmission process.
δ11,t =
A11,0 + A11,1a1,2t −1 + B11,1δ11,t −1
A21,0 + A21,1a1,t −1a2,t −1 + B21,1δ 21,t −1
δ 21,t =
(5)
δ 22,t =
A22,0 + A22,1a2,2 t −1 + B22,1δ 22,t −1
(6)
(4)
The CCC-GARCH model is relatively simple:
 δ11,t  a10  a11
δ=
 
+
 22,t  a 20  a 21
β
+  11
 β 21
β12   δ11,t −1 
β 22  δ 22,t −1 
a12   a1,2t −1 


a 22   a2,2 t −1 
(7)
α12
The coefficient which measure two prices’ relation
is constant which is not time-varying like BEKK or
DVECH. The prices are higher related when α12 is
closer to 1. This simple model can only provide the
rough result of the whole sample period.
BEKK is the most complicated model of the three.
It can capture both the time-varying correlation and
asymmetry of the information transmission. The
Diagonal VEC and CCC only have one each result.
However, the BEKK is less accurate than the other two
when focus on only one aspects. Therefore, we employ
all the three models in our research.
All the three models can be estimated in two steps.
Firstly, the GARCH parameters and then correlations are
estimated. The BHHH (Berndt, Hall, Hall and Hausman)
algorithm is usually employed in the Quasi-Maximum
Likelihood estimation.
3 Data description
Our data derives from both stock market and futures
market. The stock index used in this empirical study is
the closing price of CSI300 stock index. The CSI300
index contains 300 biggest shares from both Shanghai
and Shenzhen Stock Exchanges.
The CSI300 index futures launched in April 2010
which was relatively late. The introduction of the index
futures makes the Chinese investors could make profits
by selling out for the first time. The expiration day of the
futures is the third Friday of the contract’s delivery
month. And the delivery price is the expiration day’s last
two trading hours’ arithmetic average of index points
multiplied by RMB 300. Both the Shanghai and
Shenzhen stock exchanges trade from 9:30 a.m. to 11:30
a.m. and 1:00 p.m. to 3:00 p.m., while the CSI 300 index
futures is traded from 9:15 a.m. to 11:30 a.m. and 1:00
p.m. to 3:15p.m.
The data’s sample period is from April 19, 2010,
the beginning of the index futures, to July 16, 2014, of
about1026 trading days. The length of the time is
relatively short when compared to other researches.
However, it is the best we can do since the late
introduction of the index futures. The futures and the
underlying index prices are from Resset cooperation. The
main contract (most active) often shifts one or two days
before the delivery day. Therefore, we change the sample
contract in the third Wednesday every month, two days
before the delivery day. Fig. 1 presents the whole data's
pattern. The price of the index and futures is very close
and the basis is under 20 for most of the time, so cannot
describe clearly in Fig. 1. The trend of the graph shows
the Chinese stock market still suffered the bear market in
the last four years after the 2008 globe financial crisis.
Tab.1 summarizes the common statistics of the
daily returns. The skewness of the stock is less than the
futures, which means the futures has a fatter tail and
higher peak. The kurtosis of both markets are very high.
This means we can view the data are not normally
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I(1) processes.
The Johansen trace test is further applied. The
results indicate the existence of cointegration relation,
regardless of the existence of a linear deterministic trend
in the series. Consequently, the two series have a linear
combination that is unit-root stationary and there is a
long-run equilibrium relationship between the two prices.
We therefore expect the two markets are related and their
volatilities can be further described by multivariate
GARCH models.
distributed. The Jarque-Bera test also rejects the
hypothesis of normal distribution. Finally, the L-B test
shows the autocorrelation of series.
Tab.1 Sample statistics of the CSI index and index futures
daily return
Statistics
Stock
Futures
Mean
-0.0002
-0.00019
Maximum
0.0214
0.0391
Minimum
-0.0283
-0.0402
Standard deviation
0.0059
0.0064
Skewness
-0.1132
0.1182
Kurtosis
4.9000
8.2095
Jarque-Bera test
156.8177
1164.9
Ljung-Box Q-test
18.6999
27.9901
Tab.2 Stationarity tests on index and futures log prices
Lag length
ADF statistic
Result
1
-1.7162
non-stationary
ln St
0
0
-1.7403
-35.1838
non-stationary
stationary
ln Ft
0
-32.1066
stationary
ln St
Notes: the optimal lags are selected based on the Akaike
information criterion. The 1%, 5% and 10% critical values are
-3.4365, -2.8641 and -2.5682, respectively
4.2 Conditional variance curves
This research uses BHHH algorithm to perform the
maximum likelihood estimation in the S-Plus software.
Fig. 2 is the conditional variance curves based on BEKK
estimation. The curves based on CCC and DVEC models
which are not presented since they are similar to the
BEKK model. From the curves, we can see the futures
has more dynamics than the index in most of the time.
This is consistent with the fact that the component stocks
of the index often reach the daily limits and stop
changing, and the futures price only reached the daily
price limit once in the last five years. Therefore, the price
movement of the component stocks could be restrained
by the daily limits and futures price moves freely.
Another possible reason is that futures trades 30 minutes
longer than the spot market. Hence, the futures contains
more information than the spot market. In addition, the
variances of the two assets were higher in the futures’
infancy stage. This accords with the earlier researches
that the futures didn’t improve the price discovery
process significantly and brought disturbance to the cash
market when it’s newly launched [4,5]. The long-run
trend of the conditional variance is decreasing, which
could be a good sign that the futures market matures
gradually.
Fig. 3 shows the correlation between the CSI 300
index and the futures. As can be seen, the two series are
highly correlated when the conditional variance are
Fig.1 CSI 300 index and futures’ movement, 2010-2014
4 Empirical Result
4.1 Cointegration test
In efficient markets, arbitrageurs will pair trades to
eliminate the statistically abnormal basis of futures and
cash markets. Arbitrage trading causes information
transmission between the two markets, resulting in an
equilibrium relationship. The relationship between the
markets must be investigated before modelling them with
multivariate GARCH models. The existence of the
cointegration relation should therefore be examined. The
standard unit root procedure, the augmented
Dickey-Fuller test, is used to test the data’s stationarity
first. The results show that the null hypothesis of a unit
root is not rejected. The log prices of cash and futures are
non-stationary. However, the null hypothesis is strongly
rejected for the first-differenced series. The log prices are
Tab.3 Johansen trace tests on index and futures prices
No deterministic trend
Linear deterministic trend
H0
r=0
r≤1
ln Ft
T
C (5%)
Decision
T
C (5%)
Decision
76.4027
3.9215
20.2618
9.1645
R
F
75.5772
3.0965
15.4947
3.8414
R
F
Notes: C is the critical value. R indicates that we reject the null hypothesis. F indicates a failure to reject.
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stable and less correlated when variance has higher peaks.
Most of the time, the correlation value is close to 1. This
indicates the arbitrage opportunity is rare and the
information transmission between the two markets is
quick and efficient. Low correlation takes place in the
middle of each year. It was even under 0 for a few days
in the first year. This could be resulted in dividend
paying to the shareholders. Because it’s hard to predict
the impending dividend, the price of the two series will
disperse for weeks before the announcements. This
dispersion could be risky for arbitragers.
4.3 Dynamics spillover
Tab.4 shows the estimated result from the three
multivariate GARCH models. In CCC and BEKK
models, A12 and B12 indicate the volatility spillovers
from the index to the futures, A21 and B21 show the
volatility spillovers from the futures to the index. The
DVEC model is symmetry which cannot distinguish the
direction of volatility transmission. Therefore A12= A21
and B12= B21. Most of the estimated coefficients are
significant at 0.01 level. Whilst the A11 and A21 from the
CCC model are not significant. This means the CCC
model cannot captures the short-lived information
transmission from the futures market to the cash market.
The hypothesis that the correlation is constant will lead
the overlook of the short-run volatility transmission. It
also makes the long-run transmission result stronger and
more obvious than in the other two models. However,
from the BEKK model we can see the bidirectional
influence is significant for both long and short
dimensions. The likelihoods of three models also
indicate the BEKK fits best. Furthermore, A21 is bigger
than A12 but B21 is less than B12. This diverse result
indicates that the two market lead in short-run and
long-run respectively. The spot market contains more
long-run information. However, the futures market reacts
more quickly for short-lived news.
From each model, B11 is larger than B22, which
means that the long-run persistence of the futures is
stronger and the spot market is more risky. Also, Aii is
much smaller than Bii, which means the long-run
volatility is more important. From DVEC, A12 is much
smaller than B12, which also means long time volatility
transmission is more important.
Fig.2 Conditional variance of the index and futures
Fig.3 Conditional correlation of the index and future
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Tab.4 Estimation from three multivariate GARCH models
CCC
BEKK
C11
DVEC
coeff
sig
coeff
sig
coeff
sig
0.0014
0.6607
0.1089
0.0000
0.0300
0.0000
0.0581
0.0300
0.0299
0.0000
C21
C22
0.0015
0.5168
0.0453
0.1384
0.0363
0.0000
A11
0.0050
0.5888
0.3357
0.0000
0.1369
0.0000
A21
0.0050
0.4224
-0.2616
0.0000
0.1385
0.0000
A12
0.1000
0.0000
-0.1831
0.0005
A22
0.1000
0.0000
0.4059
0.0000
0.1454
0.0000
B11
0.9799
0.0000
0.8869
0.0000
0.7971
0.0000
B21
0.9799
0.0000
0.1010
0.0008
0.7928
0.0000
B12
0.7498
0.0000
0.0843
0.0011
22
0.7499
0.0000
0.8818
0.0000
0.7786
0.0000
B
LogL
-1179.74
-816.431
5 Conclusion
With three multivariate GARCH models, we
examine the volatility spillover between the CSI 300
index futures and its underlying spot market. The dataset
contains four years daily returns, which is the longest to
our knowledge in Chinese market. The volatility of the
futures is higher than the index because the stock index
could be restrained by daily price and trade 30 minutes
less than the futures. The correlation of the two assets is
high for most of the time and lower when shares paying
dividend. The BEKK works best for its capturing the
bidirectional information transmission between the index
and futures. The bidirectional movement is significant.
The stock market contains more long-run information.
The futures are more sensitive for the sudden news.
However the futures play a dominant role in the price
discovery process.
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