Information spillover effects between stock and option markets

Fredrik Berchtold and Lars Nordén1

School of Business, Stockholm University,

S-106 91 Stockholm, Sweden.

Abstract

This study analyses information spillover effects between the Swedish OMX stock index and

the index option market. Two types of information are analysed in a bivariate Vectorized

Autoregressive (VAR) setup with Generalized Autoregressive Conditional Heteroskedasticity

(GARCH) errors. The first type represents information where an informed investor knows

whether the stock index will increase or decrease. The second type is less specific, the

direction is unknown, but an informed investor knows that the stock index either will increase

or decrease. Possible information spillover effects are examined within a bivariate VAR-

BEKK GARCH setting, with shocks to the Swedish OMX stock index and a delta neutral

OMX options strangle portfolio as approximations of directional and undirectional

information. Significant conditional variance spillover effects are detected. Mainly, today’s

options strangle shock have an effect on tomorrow’s conditional index returns variance;

whereas stock index shocks not appears to distress the conditional option strangle variance.

This is consistent with undirectional information preceding directional information or

information spillover from the option market to the stock market.

Keywords: Information asymmetry, Spillover, Multivariate, VAR, GARCH

JEL classification: G10; G13; G14

1 Please send correspondence to Lars Nordén, e-mail: ln@fek.su.se

Information spillover effects between stock and option markets

Abstract

This study analyses information spillover effects between the Swedish OMX stock index and

the index option market. Two types of information are analysed in a bivariate Vectorized

Autoregressive (VAR) setup with Generalized Autoregressive Conditional Heteroskedasticity

(GARCH) errors. The first type represents information where an informed investor knows

whether the stock index will increase or decrease. The second type is less specific, the

direction is unknown, but an informed investor knows that the stock index either will increase

or decrease. Possible information spillover effects are examined within a bivariate VAR-

BEKK GARCH setting, with shocks to the Swedish OMX stock index and a delta neutral

OMX options strangle portfolio as approximations of directional and undirectional

information. Significant conditional variance spillover effects are detected. Mainly, today’s

options strangle shock have an effect on tomorrow’s conditional index returns variance;

whereas stock index shocks not appears to distress the conditional option strangle variance.

This is consistent with undirectional information preceding directional information or

information spillover from the option market to the stock market.

Keywords: Information asymmetry, Spillover, Multivariate, VAR, GARCH

JEL classification: G10; G13; G14

2

1. Introduction

In a review article, Madhavan (2000) suggests that asymmetric information models by

Copeland and Galai (1993), Glosten and Milgrom (1985), Kyle (1983), Easley and O’Hara

(1987), Black (1993), Foster and Viswanathan (1994) have a central role in the market

microstructure literature. In these models it is assumed that market makers, obliged to

simultaneously quote buy and sell prices of financial assets, yielding the bid-ask spread, have

an information disadvantage compared to informed investors. To protect themselves market

makers have to quote bid-ask spreads large enough to compensate for losses arising from

trading with these informed investors. The result is higher transaction costs for less informed

investors.

With the stock market in mind, two different types of information can be identified, which

implies two cases of informed investors. In one case informed investors know the direction of

the price of certain stocks, which uninformed investors do not know. In the other case,

informed investors only know that the stock prices will change, but not whether the prices

will increase or decrease. The first information type can be called directional information and

the second undirectional information. The first type of informed investors is likely to trade in

the stock market, whereas the second type, having undirectional information, is likely to trade

in the options market.

In empirical studies Cherian and Jarrow (1998) and Nandi (1999) distinguish between these

two types of information. The purpose of this study is to investigate the relationship between

these two types of information. In doing so, stock index and options strangle returns are

modelled as a bivariate Vectorized Autoregressive (VAR) process, where the variance-

covariance matrix follows a bivariate GARCH(1,1) process2 estimated with the BEKK

representation suggested by Engle and Kroner (1993).3 This setup enables an investigation of

lead-lag relationships, or information spillover effects, in the return and variance-covariance

equations.

2 GARCH is short for Generalised Autoregressive Conditional Heteroskedasticity. See e.g. Engle (1982) and Bollerslev (1986). 3 In an early version of the paper Yoshi Baba and Dennis Kraft contributed, which led to the acronym (BEKK).

3

This study contributes to previous research in several ways. First, the causal conjunction of

information asymmetry has not been empirically quantified in a similar manner before. The

BEKK model provides a framework for investigating whether directional and undirectional

information are independent or if one type of information precedes the other. Intuitively, it is

reasonable to assume that undirectional information leads directional information, as it is

more general. Secondly, both types of information are defined as stock index and options

strangle shocks. Thereby, it is possible to test informational lead-lag relationships between the

stock and options market, taking into account spillover effects in the first moment (mean

equations) and the second moment (variance-covariance equations). The lead-lag relationship

between stock and related futures markets has been extensively researched, for example by

Stoll and Whaley (1990), Chan et al. (1991) and Chan (1992), but few have studied the stock

index and index options markets.4 As a final contribution, Swedish index options data are

analysed. This is the first time anyone has used data from the Swedish stock and options

markets in this setting.

The bivariate VAR(1) – full BEKK GARCH(1,1) model is adequate for stock index and

options strangle returns. In the VAR equations, no significant autocorrelations are detected,

indicating no information spillover effects between stock index and options strangle returns or

vice versa. More importantly, significant information spillover effects between the Swedish

stock market and options market is detected in the variance-covariance equations. Somewhat

simplified, lagged squared stock index and options strangle shocks do affect the conditional

stock index variance, whereas the conditional options strangle variance only is affected by

lagged squared options strangle shocks. Likewise, the conditional covariance is significantly

affected by lagged squared strangle shocks, but not by lagged squared stock index shocks. In

all three conditional variance/covariance equations past vales of the conditional

variance/covariance also matters. These results are consistent with the idea that undirectional

information precedes directional information, or that information spills over from the option

market to the stock market.

4 Ng and Pirron (1996), Koutmos and Tucker (1996) as well as Kavussanos and Nomikos (2000) explicitly study second moment spillovers between the cash and futures markets. In addition to testing first moment spillovers, Cheung and Ng (1996) realize that volatility reflects information, and that second moment (volatility) spillovers are important as well. Also, Ross (1989) argues that volatility is related to the information flow.

4

The remainder of the study is organised as follows. Section 2 contains a description of the

Swedish market for OMX-stock index options. Section 3 presents the data and the

methodology of the study, whereas section 4 contains the results of the empirical analysis.

The study is ended in section 5 with some concluding remarks.

2. The Swedish market for OMX index options and futures

In September 1986 the Swedish exchange for options and other derivatives (OM) introduced

the OMX index, a value weighted stock index based on the 30 most actively traded stocks at

the Stockholm Stock Exchange (StSE). The purpose was to use the index as an underlying

security for trading standardised European options and futures. Since the introduction, the

trading volume has grown substantially. Presently, it is ranked among the ten largest stock

index options markets worldwide.5

All derivatives at OM are traded with a fully computerised system. The trading system

consists of an electronic limit order book hosted by OM. During trading hours investors

submit market or limit orders to either buy or sell a certain quantity of derivative contracts. If

possible, an order is matched against those already in the order book. If not, the order is

stored as another limit order. The limit order book is complemented with an “upstairs

market”. If an investor wishes to trade outside the order book he or she can phone in the order

to OM. Those orders are not added to the book. Instead, OM tries to locate a counterpart and

execute the order manually. Trades can also be executed outside the exchange. Such trades

should be reported to OM no later than fifteen minutes prior to the opening on the subsequent

trading day.

All trading in derivatives at the OM is conducted by members of the exchange.6 A member is

either a dealer or market maker. The trading environment constitutes a combination of an

electronic matching system and market making system.7 Market makers are likely to endorse

5 The largest index options markets in the world (based on trading volume in 1993) are the S&P 100 and the S&P 500 markets in the U.S. 6 The OM is the sole owner of the London Securities and Derivative Exchange (OMLX). The two exchanges are linked to each other in real time. This means that a trader at the OMLX has access to the same limit order book as a trader at the OM. In 1995, 35 members were registered at the OM and 50 at the OMLX. 7 Compare e.g. the trading system at the CBOE, which is a continuous open-outcry auction among competitive traders; floor brokers and market makers.

5

liquidity by quoting bid-ask spreads. Trading based only on a limit order book could exhibit

problems with liquidity since the high degree of transparency may adversely affect the

willingness of investors to place limit orders to the market. The trading system at StSE is

based on the same kind of limit order book as at OM. However, there are no market makers.

For the OMX index, European call and put options as well as futures contracts exist. On the

fourth Friday each month, when the exchange is open, one series of contracts expires and

another one with time to expiration equal to three months is initiated. For example, towards

the end of September, the September contracts expire and are replaced with December

contracts. At that time, the October (with time to expiration equal to one month) and the

November contracts (with a time left to expiration equal to two months) are also listed. In

addition to this maturity cycle, option and futures contracts with maturity up to two years

exist. These contracts expire in January and are included in the maturity cycle when there is

less than three months left to expiration. The maturity cycle applies for OMX index call and

put options, as well as futures.

For options a wide range of strike prices is available. Before November 28, 1997, strike prices

are set at 20 index point intervals. Thereafter, starting with contracts expiring in February

1998, strikes are set wider apart – at 40 index point intervals. On April 27, 1998, OM decided

to split the OMX index with a factor of 4:1, and to amend the regulatory framework once

again. After the split strike prices below 1,000 points are set at 10 point intervals, whereas

strike prices above 1,000 points are set at 20 point intervals. When options with new

expiration dates are introduced, strike prices are chosen so that they are centred at the current

level of the OMX index. Further, as the stock index increases or decreases considerably,

contracts with higher or lower strike prices are introduced. Thus, the range of strike prices

depends on the history of the OMX index. Actual introductions of new strikes during the

expiration cycle are reflected by the demand of the dealers and market makers.

6

3. Methodology and data

The data set consists of daily closing prices for all OMX index options contracts between

October 24, 1994, and June 29, 2001. The data, obtained from OM, includes closing bid-ask

quotes, last transaction prices, daily high and low transaction prices, number of options

contracts traded and the transacted amount in SEK as well as open interest for each contract.

The bid-ask spread represent the best bid and ask quotes in the limit order book at the close of

the exchange. Daily OMX index values, also obtained from OM, are constructed from daily

closing transactions prices of the OMX stocks.

From this data set, two daily return series are constructed, one for the OMX index and another

for a delta neutral options strangle position. The stock index return on day t ( r ) equals the

difference between the natural logarithm of the stock index closing price on day t ( ) and the

corresponding price on day t ( ):

t,1

tI

1− I 1−t

(1) 1,1 lnln −−= ttt IIr

A delta neutral options strangle position is initiated on day 1−t by buying fractions of

a call option, with the nearest strike above the stock index level, and of a put option,

with a strike just below the stock index level. The options strangle position is held until day t,

when it is closed and the return ( r ) is calculated as:

1, −tcw

1−,tpw

t,2

(2) )ln()ln( 11,11,1,1,,2 −−−−−− +−+= ttpttcttpttct PwCwPwCwr

where is the mid-quote of the call, i.e. the average of the bid-ask quotes, and the

corresponding put mid-quote on day t. The options strangle weights are obtained as

and

tC

−=

tP

)/( ∆−∆∆w )/(1,1,1,1, −−−− tptctptc 1,1,1,1, −−−− tptctctpw , where 1, −∆ tc

( ) is the estimated delta of the call (put) on day 1, −∆ tp 1−t . Delta is calculated using the

Black (1976) model, with the OMX-index futures contract as underlying security. The

∆−∆∆−=

7

implied volatility is calculated from midpoint of the call (put) quotes and simultaneous

midpoint of the quotes futures. In the implied volatility calculations, daily rates of Swedish 1-

month Treasury bills are used as a proxy for the risk-free interest rate.

To obtain a time series of options strangle returns, a new options strangle position is formed

every trading day, with the options currently closest to out-of-the-money. Furthermore, option

series closest to expiration are always used, except during expiration weeks. Each Thursday

the week before expiration, the positions are “rolled over” to the next options series. For

example, on a Thursday the week prior to the January expiration week, January options held

from Wednesday close until Thursday close are sold at the prevailing mid-quotes. Then an

options strangle position is formed using Thursday mid-quotes of February contracts, held

until Friday’s close. Thereafter, options strangle positions comprises February options until

the next rollover at the end of February. If the Friday before the expiration week is a holiday,

the rollover is initiated at Wednesday’s close.

Table 1 presents descriptive statistics for stock index and options strangle returns for the

sample period. In total, both time series have 1673 observations. The daily standard deviation

of options strangle returns is 0.0986, rather high compared to 0.0153 for stock index returns.

There are indications of fat tails, most notably for options strangle returns. Another striking

feature is the positive skewness of options strangle returns. The Ljung-Box Q(5) values, up to

five daily lags, indicate no autocorrelation in returns. On the other hand, the null hypothesis

of no autocorrelation in squared returns is strongly rejected for stock index and options

strangle returns since the corresponding Q(5) values have p-values < 0.0001. The reported

cross correlation coefficient for stock index and option strangle returns is low for raw returns,

and somewhat higher for squared returns. Although crude diagnostics, auto- and cross

correlation in squared returns indicate that the bivariate GARCH model is an appropriate

framework for analysing information spillover effects between the stock index and options

markets.

The returns ( and )tr ,1 tr ,2 in equation (1) and (2) are stationary. As stated by Sims (1972), the

VAR specification can handle an infinite number of stationary variables. In the bivariate

VAR(1) case, with two variables, the model is:

8

(3)

+

+

=

−

−

t

t

t

t

t

trr

rr

,2

,1

1,2

1,1

2,21,2

2,11,1

2

1

,2

,1εε

φφφφ

µµ

where the residuals ( t,1ε and t,2ε ) are interpreted as unexpected stock index and options

strangle returns, or shocks, at time t. The tε vector, which contains t,1ε and t,2ε , is assumed

to be normally distributed with the conditional variance-covariance matrix . tH

The parameters in the conditional variance-covariance matrix can be modelled in several

ways. One way is to model it as a bivariate GARCH(1,1) process, following Engle and

Kroner’s (1993) vec-representation:

(4) 11110 −− ++= ttt hGACh η

where , , C is a )( tt Hvech = )( ttt vec εεη ′= 0 122 × parameter vector, and G are 1A 122 22 ×

parameter matrices. Writing the model in this way, the covariance terms appear twice. There

is one equation for and another for h as all off-diagonal terms appear twice in each

equation, i.e.

t,12

1,2 −t

h t,21

1,1 −t εε and 1,11 t,2 −t −εε , as well as and appear in each

equation. Therefore, according to Engle and Kroner (1993), it is possible to omit the

equation and to omit all coefficients for

1,12 −th 1−t,21h

th ,21

1,11 −t,2 −t εε and , as they are redundant. After

omitting the redundant terms, the model can be written as:

1,21 −th

(5)

+

+

=

=

−

−

−

−

−−

−

1,22

1,12

1,11

333231

232221

131211

21,2

1,21,1

21,1

333231

232221

131211

03

02

01

,22

,12

,11

t

t

t

t

tt

t

t

t

t

thhh

ggggggggg

aaaaaaaaa

ccc

hhh

h

ε

εε

ε

where each and G matrix contain nine free parameters. 1A 1

For empirical purposes, the vec-representation in equation (5) contains 21 parameters and can

be difficult to estimate. Engle and Kroner (1993) proposed the BEKK-representation, which

9

imposes restrictions on the parameters in equation (5) within and across equations. Using the

notation in Engle and Kroner (1993), the parameterization becomes:

(6) *11

*'1

*111

*'1

*0

*'0 GHGAACCH tttt −−− +′+= εε

where C , and are matrices and C is triangular. The BEKK-representation can

also be written in the following form:

*0

*1A *

1G 22× *0

(7)

+=

−−−

−−−*22

*21

*12

*11

21,21,21,1

1,21,12

1,1'

*22

*21

*12

*11*

0*'0

aaaa

aaaaCCH

ttt

tttt

εεε

εεε

+ − *

22*21

*12

*11

1

'

*22

*21

*12

*11

ggggH

gggg

t

or:

(8) 21,2

2*211,21,1

*21

*11

21,1

2*1111,11 2 −−−− +++= ttttt aaaach εεεε

1,222*

211,12*21

*111,11

2*11 2 −−− +++ ttt hghgghg

21,2

*22

*211,21,1

*22

*11

*12

*21

21,1

*12

*1112,12 )( −−−− ++++= ttttt aaaaaaaach εεεε

1,22*22

*211,12

*22

*11

*12

*211,11

*12

*11 )( −−− ++++ ttt hgghgggghgg

21,2

2*221,21,1

*22

*12

21,1

2*1213,22 2 −−−− +++= ttttt aaaach εεεε

1,222*

221,12*22

*121,11

2*12 2 −−− +++ ttt hghgghg

10

Compared to the vec-representation in equation (5), the BEKK-representation in equation (7)

or (8), hereafter called full BEKK GARCH(1,1), contains 11 parameters instead of 21. The

two representations are equivalent when the following non-linear restrictions are applied on

the matrices and : * *

A G

* *

****

1A 1G

(9) , a , a 2*1111 aa = *

21*1112 2 aa= 2*

2113 a=

, a , a *12

*1121 aaa = )( *

22*11

*12

*2122 aaaa += *

22*2123 aa=

, a , a 2*1231 aa = *

22*1232 2 aa= 2*

2233 a=

2*1111 gg = , , *

21*1112 2 ggg = 2*

2113 gg =

*12

*1121 ggg = , , )( *

22*11

*12

*2122 ggggg += *

22*2123 ggg =

2*1231 gg = , , *

22*1232 2 ggg = 2*

2233 gg =

As a further restriction of the parameterization, the diagonal representation suggested by

Bollerslev et al. (1988) can be used. It restricts the off-diagonal elements in and to

zero. Consequently, each conditional variance only depends on past values of itself and its

own lagged squared residuals, whereas the conditional covariance depends on past values of

itself and the lagged cross-product of residuals. In the diagonal BEKK-representation of

equation (7), off-diagonal terms in and G are restricted to zero, i.e.

. Hence, equation (8) simplifies to:

1 1

1A 1

021122112 ==== ggaa

(10) ++= −2

1,12*

1111,11 tt ach ε 1,112*

11 −thg

1,21,1*22

*1112,12 −−+= ttt aach εε 1,12

*22

*11 −+ thgg

11

21,2

2*2213,22 −+= tt ach ε 1,22

2*22 −+ thg

In comparison to the diagonal vec-representation the following non-linear restrictions are

applied to the parameters in equation (10):

(11) , a , a 2*1111 aa = *

22*1122 aa= 2*

2233 a=

2*1111 gg = , , *

22*1122 ggg = 2*

2233 gg =

As a result, estimating the diagonal model using the BEKK-representation, hereafter called

the diagonal BEKK GARCH(1,1), involves only seven parameters (including three constant

terms) rather than nine, which have to be estimated using the diagonal vec-representation.

In this study, the full BEKK GARCH(1,1) model in equation (7) or (8) is estimated, together

with the VAR(1) model in equation (3), in order to evaluate the possibility of information

spillover effects between the stock and the options market. The actual estimation is carried

out in two steps. First, the VAR model is estimated to obtain the vector tε , which contains

the stock index and options strangle shocks t,1ε and t,2ε . Thereafter, this vector is used as

input in the estimation of the GARCH model. The full BEKK GARCH(1,1) model has an

equation for , the conditional stock index variance, , the conditional options strangle

variance, and , which is the conditional covariance between stock index and option

strangle returns. This model allows for spillover effects. For instance, in the first equation,

is a function of lagged squared stock index shocks ( ) as well as lagged squared

options strangle shocks ( ) and the cross term

t,11

h12

h

2

2

t,

1ε

1−

h22

,1 t

t,

th ,11 1, −t

1,2 −t1,2 −tε εε . Also, lagged conditional

variance/covariance terms from the other equations are included. Similar cross-equation terms

are included in the second equation (for ) and third equation (for ). Clearly, if there

are significant spillover effects the parameters associated with the cross-equation terms

should be non-zero. Consequently, an overall test of the presence of

spillover effects can be performed simply by testing the null hypothesis

t,12h h

*

t,22

*g2121a = *12g =*

12a =

12

021122112 ==== ggaa

11

21,2 −tε

****

h

*

2

, i.e. to compare the full BEKK GARCH model with the diagonal

BEKK GARCH model in equation (10).

4. Empirical results

Table 2 presents results from the estimation of the bivariate VAR(1) model of stock index and

options strangle returns, that is equation (3). As can be seen from the p-values, no

autocorrelation coefficient is significant at reasonable levels. Hence, no spillover effects are

found in the mean equations.

The full BEKK GARCH(1,1) coefficients in equation (8) are provided in Table 3. The model

is estimated assuming joint normally distributed errors. There is strong evidence of

heteroskedasticity in stock index and options strangle returns. The full BEKK GARCH(1,1)

model captures this conditional stock index and options strangle variance well, as the Ljung-

Box Q(5)-test indicates no remaining autocorrelation in squared stock index residuals at the

five percent level. A question mark has to be set for squared options strangle residuals, as the

Ljung-Box Q(5)-test indicates remaining autocorrelation at the five percent level.8 Also, the

model reduces the kurtosis in returns considerably, compared with the results from Table 1.

Especially for stock returns, the Jarque-Bera test can not reject the null hypothesis of

normality in the standardised residuals. However, it is still possible to reject the

corresponding normality hypothesis for the standardised options strangle residuals, as the

Jarque-Bera test statistic is very large.

The p-values in Table 3 indicates that most coefficients in the full BEKK GARCH(1,1) model

are significant. In the conditional stock index variance equation, in equation (8), the

constant term ( c ) is significant, as are the coefficients a and . This implies that both

lagged squared stock index shocks ( ) and lagged squared options strangle shocks

( ) affect the conditional stock index variance. Also, the lagged cross term

t,11

*2111 a

1,1 −tε

1,21,1 −− tt εε

8 The residuals correspond to the return shocks from the VAR(1) model, standardized with respect to the estimated . tH

13

affect the conditional stock index variance, as both coefficients and are significant.

Further, since the coefficient is significant, whereas is not, is a function of

itself lagged one period, but not of the lagged conditional option strangle variance and

conditional covariance.

* *

* *

*

2

2

*

h

*

** *

*

*

11a 21a

t,11

2,1 tε

11g 21g

*12a

)22

h

tIn the conditional variance equation for option strangle returns, h in equation (8), the

coefficient for the constant term c is not significant. Further, the coefficient a is

significant whereas the coefficient is not. This implies that lagged options strangle return

shocks ( ) affect the conditional variance of option strangle returns but that lagged stock

index shocks ( ) do not. Also, it is doubtful whether the lagged cross term

affects , since its presence in the equation depends on the multiplication of one

coefficient ( ), which is significant, and another ( a ), which is not. Also, it is only the

own lagged conditional variance which affects significantly, and not the lagged

conditional variance of stock index returns or the lagged conditional covariance, since the

coefficient is significant whereas is not.

,22

13

*12a

g

22

,21− tε

1,2 −tε

th ,22

*22a

*22g

1,1 −tε 1,1 −tε

12

t,22

12

In the equation for the conditional covariance, h in equation (8), the coefficient for the

constant term is not significant. Furthermore, is related to through the

multiplicative term a , where is significant and is not. Hence, it is doubtful if

squared lagged stock index return shocks affect the conditional covariance. On the other hand,

both coefficients in the multiplicative term are significant. This implies that lagged

squared options strangle return shocks affect the conditional covariance. The lagged cross

term also has some meaning for the conditional covariance, since the second

multiplicative term in the expression consists of two significant

coefficients. Finally, the lagged conditional covariance has a significant contribution since the

two coefficients in second term in the expression ( ) are significant.

t,12

*22a

*12a +

*21g

12c

1−

th ,12

**11aa

*12g +

1−

1211a 11a

21a

( *21a

,21,1 − tt εε

22g*11g

14

Summing up, the specification of a bivariate VAR(1) – full BEKK GARCH(1,1) model is

adequate for stock index and options strangle returns. An overall LR-test of spillover effects,

testing the null hypothesis , i.e. comparing the full and diagonal

models, supports the full model with a p-value 0.003 (4 d.f.).

021122112 ==== ggaa ****

Also, the estimation, following a two step procedure, converged without problems. In the

VAR model, no significant autocorrelation is detected, indicating no spillover effects between

stock index and options strangle returns. More importantly, the p-values of the coefficients in

the conditional variance-covariance equations indicate significant second moment spillover

effects between stock index and options strangle returns. Somewhat simplified, lagged

squared stock index and options strangle return shocks do affect the conditional stock index

variance. In total, there is a complex structure where some cross terms also matters. Past

values of the conditional stock index variance also matters. No simple spillover effect is

identified from stock index shocks to the conditional options strangle variance.

Engle and Ng (1993) present "news impact curves", figures displaying the response of the

conditional variance to new information, the information shocks or “news”. The implied

relation between lagged squared information shocks and the current level of the conditional

variance is plotted, while holding earlier information constant. In particular, the lagged

conditional variances and covariance are evaluated at the respective unconditional level. For

the full BEKK GARCH(1,1) model in equation (8), each conditional variance/covariance is

allowed to be a function of the two types of information shocks. Thus, news impact surfaces

are given by:

(12) 21,21,21,1

21,1,11 0.00000038.00.10580013.0 −−−− +−+= ttttth εεεε

221211 0000.00079.08917.0 σσσ +++

21,21,21,1

21,1,12 0.00050279.00.00670170.0 −−−− +++−= ttttth εεεε

221211 0041.09150.01049.0 σσσ ++−

15

21,21,21,1

21,1,22 0.00740035.00.00040135.0 −−−− +++= ttttth εεεε

221211 9399.02154.00123.0 σσσ +−+

where 11σ is the unconditional variance of stock index returns, 22σ the unconditional

variance of option strangle returns and 12σ the corresponding unconditional covariance.

Figure 1, 2 and 3 display news impact surfaces for equation (12), the full BEKK GARCH(1,1)

model. As can be seen in Figure 1, the response of the conditional stock index variance to

stock index return shocks is clearly positive while the response to option strangle shocks

appears negligible. Here it should be noted that the standard deviation of option strangle

returns is very high. Positive or negative stock index shocks (as they are squared) tend to

increase the conditional stock index variance. Figure 2 reveals positive responses of the

conditional options strangle variance to stock index and options strangle shocks, as the

surface is bowled. In this case, the magnitude of stock index shocks is exaggerated because of

the equal scaling of the axis.

From Figure 3, it is seen that the response of the conditional covariance is more complex. The

response is stronger for stock index shocks, but options strangle shocks clearly change the

conditional covariance as well. In economic terms, given the comparatively high standard

deviation of options strangle returns; this may very well dominate.

The overall economic implications of information spillover effects from options strangle

shocks to the conditional stock index variance is that today’s options strangle shock contains

significant information about tomorrow’s conditional stock index variance. No significant

spillover effect is identified from stock index shocks to the conditional options strangle

variance. Again, the figures should be interpreted carefully as the standard deviation of

options strangle returns is more than five times higher than the standard deviation of stock

index returns.

5. Concluding remarks

16

In this study information spillover effects from the options market to the stock market are

identified in an information asymmetry context. Two types of information and two types of

investors are identified. The first type of informed investor knows if the stock index will

increase or decrease tomorrow. The second type has undirectional information. It is assumed

that the first type profits from trading stocks and the second from trading index options. Here,

two types of information is quantified in a bivariate VAR(1)-full BEKK GARCH(1,1) setup.

This allows a decomposition of the returns into conditional variance and standardized

residuals, where the latter also are called information shocks or news. Tomorrow’s stock

index shock is new information; the future direction of the stock index is revealed. Similarly,

tomorrows options strangle shocks are new undirectional information.

The results are significant information spillover effects in the conditional stock index variance

equation. Information spills over from options strangle shocks to the conditional stock index

variance. In other words, squared lagged options strangle shocks do affects the conditional

stock index variance. Undirectional information about the stock index is revealed through the

trading of index options, consistent with undirectional information preceding directional

information.

Information asymmetry and information spillover effects are linked to the market

microstructure literature in several ways, for example to the bid-ask spread, i.e. market

efficiency and arbitrage possibilities. It is also possible that information lead-lag relationships

are important for price discovery. If the stock market leads the options market, or vice versa,

then information from one market says something about subsequent prices movements or the

volatility changes in the other market. Volatility spillover effects might imply that volatility in

one market is transmitted to the market, another possible area for price discovery.

Information spillover effects are useful for decisions regarding market making in stocks and

options, investment strategies involving price discoveries and risk management practices such

as hedging and value at risk calculations. An open question is if the options market

destabilises the stock market. Here the leverage could exacerbate the volatility in the

underlying stocks. Increased volatility would reduce investor’s confidence, possibly leading

to higher trading costs, lower liquidity and thereby implicitly higher cost of owning stocks, as

noted by Harris (1989).

17

18

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21

Table 1: Summary statistics for the OMX stock index and OMX options strangle returns

Statistics Index returns Strangle returns

Mean 6.76e-4 -3.19e-3

Standard deviation 0.0153 0.0986

Skewness 0.0597 1.1650

Kurtosis 6.0803 8.2054

Jarque-Bera 662.81 2,268.6

p-value of Jarque-Bera < 0.0001 < 0.0001

Raw returns correlations

Rho (lag = 1) 0.005 0.016

Rho (lag = 2) -0.029 0.032

Ljung-Box Q(5) 4.128 6.934

p-value of Ljung-Box 0.531 0.226

Squared returns correlations

Rho (lag = 1) 0.184 0.110

Rho (lag = 2) 0.255 0.047

Ljung-Box Q(5) 264.9 25.23

p-value of Ljung-Box < 0.001 < 0.001

Cross-correlations between Index returns and Strangle returns

Raw returns Squared returns

Rho (lag = -2) -0.009 0.067

Rho (lag = -1) 0.013 0.097

Rho (lag = 0) -0.120 0.513

Rho (lag = +1) 0.003 0.062

Rho (lag = +2) -0.008 0.088

22

Table 2: Results from the Bivariate VAR(1) model for stock index and options strangle

returns.

Bivariate VAR(1)

Coefficient Estimate t-value p-value

11φ 0.00545 0.2211 (0.8258)

12φ 0.00063 0.1643 (0.8694)

1µ 0.00068 1.8122 (0.0701)

21φ 0.09493 0.5978 (0.5500)

22φ 0.01754 0.7118 (0.4767)

2µ -0.0032 -1.3232 (0.1859)

23

Table 3: Results from the Diagonal and full BEKK GARCH(1,1) models for stock index and

options strangle returns.

Full BEKK GARCH(1,1) Diagonal BEKK GARCH(1,1)

Coefficient Estimate t-value p-value Estimate t-value p-value

11c 0.0013 2.9465 (0.0032) 0.0014 8.8439 (0.0000)

12c -0.0170 -0.9839 (0.3270) -0.0275 -1.3254 (0.1856)

13c 0.0135 0.6060 (0.5461) 0.0734 9.3294 (0.0000)

*11a 0.3252 21.797 (0.0000) 0.2879 29.490 (0.0000)

*12a 0.0206 0.0706 (0.9538) - - -

*21a -0.0059 -2.1433 (0.0324) - - -

*22a 0.0861 2.6796 (0.0074) 0.4398 5.0872 (0.0000)

*11g 0.9443 455.76 (0.0000) 0.9548 6996.4 (0.0000)

*12g -0.1111 -0.9989 (0.3184) - - -

*21g 0.0042 1.5418 (0.1232) - - -

*22g 0.9695 2226.2 (0.0000) 0.4412 3.6878 (0.0000)

Residuals

Stock shocks Strangle shocks Stock chocks Strangle returns

Skewness 0.1662 1.1097 0.1242 1.1344

Kurtosis 3.3608 7.7413 3.5631 8.5310

Jarque-Bera 16.774 1910.4 26.404 2614.2

Raw Q(5) 2.6809 5.2549 3.0430 5.5669

Squared Q(5) 4.4124 20.129 4.5867 1.2281

24

Figure 1: News impact surface for the conditional stock index variance, full BEKK

GARCH(1,1) model.

25

Figure 2: News impact surface for the conditional options strangle variance, full BEKK

GARCH(1,1) model.

26

Figure 3: News impact surface for the conditional covariance, full BEKK GARCH(1,1)

model.

27