Information spillover effects between stock and option markets




                                Fredrik Berchtold and L...
Information spillover effects between stock and option markets




                                          Abstract

Thi...
1. Introduction


In a review article, Madhavan (2000) suggests that asymmetric information models by
Copeland and Galai (...
This study contributes to previous research in several ways. First, the causal conjunction of
information asymmetry has no...
The remainder of the study is organised as follows. Section 2 contains a description of the
Swedish market for OMX-stock i...
liquidity by quoting bid-ask spreads. Trading based only on a limit order book could exhibit
problems with liquidity since...
3. Methodology and data


The data set consists of daily closing prices for all OMX index options contracts between
Octobe...
implied volatility is calculated from midpoint of the call (put) quotes and simultaneous
midpoint of the quotes futures. I...
 r1,t   µ1  φ1,1 φ1,2   r1,t −1   ε1,t 
(3)           r  =   + φ                      +      
        ...
imposes restrictions on the parameters in equation (5) within and across equations. Using the
notation in Engle and Kroner...
Compared to the vec-representation in equation (5), the BEKK-representation in equation (7)
or (8), hereafter called full ...
*2 2           *2
              h22,t = c13 + a22ε 2,t −1 + g 22 h22,t −1



In comparison to the diagonal vec-representat...
*     *     *      *
a12 = a21 = g12 = g 21 = 0 , i.e. to compare the full BEKK GARCH model with the diagonal
BEKK GARCH m...
*        *
affect the conditional stock index variance, as both coefficients a11 and a 21 are significant.
               ...
Summing up, the specification of a bivariate VAR(1) – full BEKK GARCH(1,1) model is
adequate for stock index and options s...
2                                        2
                 h22,t = 0.0135 + 0.0004ε1,t −1 + 0.0035ε1,t −1ε 2,t −1 + 0.007...
In this study information spillover effects from the options market to the stock market are
identified in an information a...
18
References


Bae, K. and A. Karolyi, 1994, Good news, bad news and international spillovers of return
volatility between J...
Cox, J. and M. Rubinstein, 1985, Option markets, New Jersey: Prentice Hall.


Easley, D. and M. O’Hara, 1987, Price, trade...
Koutmos, G. and M. Tucker, 1996, Temporal relationships and dynamic interactions between
spot and futures stock markets, J...
Table 1: Summary statistics for the OMX stock index and OMX options strangle returns

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

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


       ...
Figure 1: News impact surface for the conditional stock index variance, full BEKK
GARCH(1,1) model.




                  ...
Figure 2: News impact surface for the conditional options strangle variance, full BEKK
GARCH(1,1) model.




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




                            ...
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Information spillovers between stock and options markets

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Information spillovers between stock and options markets

  1. 1. 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
  2. 2. 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
  3. 3. 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
  4. 4. 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
  5. 5. 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
  6. 6. 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
  7. 7. 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 ( r1,t ) equals the difference between the natural logarithm of the stock index closing price on day t ( I t ) and the corresponding price on day t − 1 ( I t −1 ): (1) r1,t = ln I t − ln I t −1 A delta neutral options strangle position is initiated on day t − 1 by buying wc,t −1 fractions of a call option, with the nearest strike above the stock index level, and w p,t −1 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 ( r2,t ) is calculated as: (2) r2,t = ln(wc,t −1Ct + w p,t −1Pt ) − ln( wc,t −1Ct −1 + w p,t −1Pt −1) where Ct is the mid-quote of the call, i.e. the average of the bid-ask quotes, and Pt the corresponding put mid-quote on day t. The options strangle weights are obtained as wc,t −1 = − ∆ p ,t −1 /( ∆ c,t −1 − ∆ p,t −1) and w p,t −1 = − ∆ c,t −1 /(∆ c,t −1 − ∆ p,t −1 ) , where ∆ c,t −1 ( ∆ p ,t −1 ) is the estimated delta of the call (put) on day t − 1 . Delta is calculated using the Black (1976) model, with the OMX-index futures contract as underlying security. The 7
  8. 8. 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 ( r1,t and r2,t ) 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
  9. 9.  r1,t   µ1  φ1,1 φ1,2   r1,t −1   ε1,t  (3) r  =   + φ  +   2,t   µ 2   2,1 φ 2,2  r2,t −1  ε 2,t  where the residuals ( ε1,t and ε 2,t ) are interpreted as unexpected stock index and options strangle returns, or shocks, at time t. The ε t vector, which contains ε1,t and ε 2,t , is assumed to be normally distributed with the conditional variance-covariance matrix H t . 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) ht = C0 + A1ηt −1 + G1ht −1 where ht = vec( H t ) , ηt = vec(ε t ε t′ ) , C0 is a 22 × 1 parameter vector, A1 and G1 are 22 × 22 parameter matrices. Writing the model in this way, the covariance terms appear twice. There is one equation for h12,t and another for h21,t as all off-diagonal terms appear twice in each equation, i.e. ε1,t −1ε 2,t −1 and ε 2,t −1ε1,t −1 , as well as h12,t −1 and h21,t −1 appear in each equation. Therefore, according to Engle and Kroner (1993), it is possible to omit the h21,t equation and to omit all coefficients for ε 2,t −1ε1,t −1 and h21,t −1 , as they are redundant. After omitting the redundant terms, the model can be written as: a13   ε1,t −1   g11 2  h11,t   c01   a11 a12 g12 g13   h11,t −1        (5) ht =  h12,t  = c02  + a21 a22    a23  ε1,t −1ε 2,t −1  +  g 21  g 22 g 23   h12,t −1     h22,t  c03   a31 a32     a33   ε 2  g g32 g33  h22,t −1   2,t −1   31    where each A1 and G1 matrix contain nine free parameters. 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
  10. 10. imposes restrictions on the parameters in equation (5) within and across equations. Using the notation in Engle and Kroner (1993), the parameterization becomes: *' * *' * *' * (6) H t = C0 C0 + A1 ε t −1ε t′−1 A1 + G1 H t −1G1 * * * * where C0 , A1 and G1 are 2 × 2 matrices and C0 is triangular. The BEKK-representation can also be written in the following form: * ' 2 ε1,t −1ε 2,t −1  *' *  a * a12   ε1,t −1   a11 a12  * * (7) H t = C0 C0 +  11 * *   * *  a22  ε1,t −1ε 2,t −1 2 a21   ε 2,t −1   a21 a22    '  g* g12  *  g* g12  * +  11* *  H t −1  11* *   g 21  g 22    g 21  g 22   or: *2 2 * * *2 2 (8) h11,t = c11 + a11 ε 1,t −1 + 2a11 a 21ε 1,t −1ε 2,t −1 + a 21 ε 2,t −1 *2 * * *2 + g11 h11, t −1 + 2 g11 g 21h12, t −1 + g 21 h22, t −1 * * 2 * * * * * * 2 h12,t = c12 + a11a12 ε 1,t −1 + (a 21a12 + a11a 22 )ε 1,t −1ε 2,t −1 + a 21a 22 ε 2,t −1 * * * * * * * * + g11g12h11,t −1 + ( g 21g12 + g11g 22 )h12,t −1 + g 21g 22h22,t −1 *2 2 * * *2 2 h22, t = c13 + a12 ε1, t −1 + 2a12 a 22ε1, t −1ε 2, t −1 + a 22 ε 2, t −1 *2 * * *2 + g12 h11,t −1 + 2 g12 g 22h12,t −1 + g 22h22,t −1 10
  11. 11. 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 A1 and G1 : *2 * * *2 (9) a11 = a11 , a12 = 2a11a21 , a13 = a21 * * * * * * * * a21 = a11a12 , a22 = (a21a12 + a11a22 ) , a23 = a21a22 *2 * * *2 a31 = a12 , a32 = 2a12a22 , a33 = a22 *2 * * *2 g11 = g11 , g12 = 2 g11g 21 , g13 = g 21 * * * * * * * * g 21 = g11g12 , g 22 = ( g 21g12 + g11g 22 ) , g 23 = g 21g 22 *2 * * *2 g31 = g12 , g32 = 2 g12 g 22 , g33 = g 22 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 A1 and G1 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 A1 and G1 are restricted to zero, i.e. * * * * a12 = a21 = g12 = g 21 = 0 . Hence, equation (8) simplifies to: *2 2 *2 (10) h11,t = c11 + a11 ε1,t −1 + g11 h11,t −1 * * * * h12,t = c12 + a11a22ε1,t −1ε 2,t −1 + g11g 22 h12,t −1 11
  12. 12. *2 2 *2 h22,t = c13 + a22ε 2,t −1 + g 22 h22,t −1 In comparison to the diagonal vec-representation the following non-linear restrictions are applied to the parameters in equation (10): *2 * * *2 (11) a11 = a11 , a22 = a11a22 , a33 = a22 *2 * * *2 g11 = g11 , g 22 = g11g 22 , g33 = g 22 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 ε t , which contains the stock index and options strangle shocks ε1,t and ε 2,t . 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 h11,t , the conditional stock index variance, h22,t , the conditional options strangle variance, and h12,t , which is the conditional covariance between stock index and option strangle returns. This model allows for spillover effects. For instance, in the first equation, 2 h11,t is a function of lagged squared stock index shocks ( ε1,t −1 ) as well as lagged squared 2 options strangle shocks ( ε 2,t −1 ) and the cross term ε1,t −1ε 2,t −1 . Also, lagged conditional variance/covariance terms from the other equations are included. Similar cross-equation terms are included in the second equation (for h12,t ) and third equation (for h22,t ). Clearly, if there are significant spillover effects the parameters associated with the cross-equation terms * * * * a12 = a21 = g12 = g 21 should be non-zero. Consequently, an overall test of the presence of spillover effects can be performed simply by testing the null hypothesis 12
  13. 13. * * * * a12 = a21 = g12 = g 21 = 0 , 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, h11,t in equation (8), the * * constant term ( c11 ) is significant, as are the coefficients a11 and a 21 . This implies that both 2 lagged squared stock index shocks ( ε1,t −1 ) and lagged squared options strangle shocks 2 ( ε 2, t −1 ) affect the conditional stock index variance. Also, the lagged cross term ε1,t −1ε 2,t −1 8 The residuals correspond to the return shocks from the VAR(1) model, standardized with respect to the estimated H t . 13
  14. 14. * * affect the conditional stock index variance, as both coefficients a11 and a 21 are significant. * * Further, since the coefficient g11 is significant, whereas g 21 is not, h11,t is a function of itself lagged one period, but not of the lagged conditional option strangle variance and conditional covariance. In the conditional variance equation for option strangle returns, h22, t in equation (8), the * coefficient for the constant term c13 is not significant. Further, the coefficient a 22 is * significant whereas the coefficient a12 is not. This implies that lagged options strangle return 2 shocks ( ε 2, t −1 ) affect the conditional variance of option strangle returns but that lagged stock 2 index shocks ( ε1,t −1 ) do not. Also, it is doubtful whether the lagged cross term ε1,t −1ε 2,t −1 affects h22, t , since its presence in the equation depends on the multiplication of one * * coefficient ( a 22 ), which is significant, and another ( a12 ), which is not. Also, it is only the own lagged conditional variance which affects h22, t significantly, and not the lagged conditional variance of stock index returns or the lagged conditional covariance, since the * * coefficient g 22 is significant whereas g12 is not. In the equation for the conditional covariance, h12, t in equation (8), the coefficient for the 2 constant term c12 is not significant. Furthermore, h12, t is related to ε1,t −1 through the * * * * multiplicative term a11a12 , where a11 is significant and a12 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 a21a22 are significant. This implies that lagged squared options strangle return shocks affect the conditional covariance. The lagged cross term ε1,t −1ε 2,t −1 also has some meaning for the conditional covariance, since the second * * * * multiplicative term in the expression (a21a12 + a11a22 ) consists of two significant coefficients. Finally, the lagged conditional covariance has a significant contribution since the * * * * two coefficients in second term in the expression ( g 21g12 + g11g 22 ) are significant. 14
  15. 15. 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 a12 = a21 = g12 = g 21 = 0 , i.e. comparing the full and diagonal models, supports the full model with a p-value 0.003 (4 d.f.). 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: 2 2 (12) h11,t = 0.0013 + 0.1058ε1,t −1 − 0.0038ε1,t −1ε 2,t −1 + 0.0000ε 2,t −1 + 0.8917σ 11 + 0.0079σ 12 + 0.0000σ 22 2 2 h12,t = −0.0170 + 0.0067ε1,t −1 + 0.0279ε1,t −1ε 2,t −1 + 0.0005ε 2,t −1 − 0.1049σ 11 + 0.9150σ 12 + 0.0041σ 22 15
  16. 16. 2 2 h22,t = 0.0135 + 0.0004ε1,t −1 + 0.0035ε1,t −1ε 2,t −1 + 0.0074ε 2,t −1 + 0.0123σ 11 − 0.2154σ 12 + 0.9399σ 22 where σ 11 is the unconditional variance of stock index returns, σ 22 the unconditional variance of option strangle returns and σ 12 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
  17. 17. 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. 18. 18
  19. 19. References Bae, K. and A. Karolyi, 1994, Good news, bad news and international spillovers of return volatility between Japan and the U.S., Pacific Basin Finance Journal 2, 405-438. Black F., 1976, Studies of stock market volatility changes, Proceedings of the American Statistical Association, Business and Economics Statistics Section, 177-181. Black, F., 1993, Estimating expected return, Financial Analysts Journal, 36-38. Bollerslev, T., 1986, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, 307-327. Bollerslev, T., R. Engle, and J. Wooldridge, 1988, A capital asset pricing model with time varying covariances, Journal of Political Economy 96, 116-131. Chan, K., 1992, A further analysis of the lead-lag relationship between the cash market and stock index futures market, Review of Financial Studies 5, 123-152. Chan, K., Chan, K. C. and Karolyi, G. A., 1991, Intraday volatility in the stock index and stock index futures markets, Review of Financial Studies, 4, 657-684. Cherian, J. and R. Jarrow, 1998, Option markets, self-fulfilling prophecies, and implied volatilities, Review of Derivatives Research 2, 5-37. Cheung, Y. and L. Ng, 1996, A causality-in-variance test and its implications to financial markets prices, Journal of Econometrics 72, 33-48. Copeland, T. and D. Galai, 1983, Information effects on the bid-ask spread, Journal of Finance 38, 1457-1469. 19
  20. 20. Cox, J. and M. Rubinstein, 1985, Option markets, New Jersey: Prentice Hall. Easley, D. and M. O’Hara, 1987, Price, trade size, and information in securities markets, Journal of Financial Economics 19, 69-90. Engle, R. , 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation, Econometrica 50, 987-1008. Engle, R. and K. Kroner, 1993, Multivariate simultaneous generalized ARCH, Discussion Paper 89-57R, University of California, San Diego. Engle, R. F. and V. K. Ng, 1993, “Measuring and testing the impact of news on volatility,” Journal of Finance, 48, 1749-1778. Foster, F. and S. Viswanathan, 1994, Strategic trading with asymmetrically informed investors and longed-lived information, Journal of Financial and Quantitative Analysis 29, 499-518. Glosten, L. and P. Milgrom, 1985, Bid, ask and transaction prices in a specialist market with heterogeneously informed traders, Journal of Financial Economics 14, 71-100. Harris, L., 1989, The October 1987 S&P 500 Stock-Futures Basis, Journal of Finance, 44, 77- 99. Karolyi, A., 1995, A multivariate GARCH model of international transmissions of stock returns and volatility: the case of the United States and Canada, Journal of Business and Economic Statistics 13, 11-25. Kavussanos, M. and N. Nomikos, 2000, Hedging in the freight futures market, Journal of Derivatives 8, 41-58. 20
  21. 21. Koutmos, G. and M. Tucker, 1996, Temporal relationships and dynamic interactions between spot and futures stock markets, Journal of Futures Markets 16, 55-69. Kyle, A., 1983, Continuous auctions and insider trading, Econometrica 53, 1315-1335. Madhavan, A., 2000, Market microstructure: A survey. Journal of Financial Markets 3, 205– 258. Nandi, S., 1999, Asymmetric information about volatility: how does it affect implied volatility, option prices and market liquidity? Review of Derivatives Research 3, 215-236. Ng, V. and S. Pirron, 1996, Price dynamics in refinery petroleum spot and futures markets, Journal of Empirical Finance 2, 359-388. Ross, S., 1989, Information and volatility: The no-arbitrage martingale approach to timing and resolution irrelevancy, Journal of Finance 44, 1-17. Sims, C., 1972, The role of approximate prior restrictions in distributed lag estimation, Journal of American Statistical Association 67, 169-175. Stoll, H. and R. Whaley, 1990, The dynamics of stock index and stock index futures returns, Journal of Financial and Quantitative Analysis 25, 441-468. 21
  22. 22. 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
  23. 23. 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 φ11 0.00545 0.2211 (0.8258) φ12 0.00063 0.1643 (0.8694) µ1 0.00068 1.8122 (0.0701) φ 21 0.09493 0.5978 (0.5500) φ 22 0.01754 0.7118 (0.4767) µ2 -0.0032 -1.3232 (0.1859) 23
  24. 24. 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 c11 0.0013 2.9465 (0.0032) 0.0014 8.8439 (0.0000) c12 -0.0170 -0.9839 (0.3270) -0.0275 -1.3254 (0.1856) c13 0.0135 0.6060 (0.5461) 0.0734 9.3294 (0.0000) * a11 0.3252 21.797 (0.0000) 0.2879 29.490 (0.0000) * a12 0.0206 0.0706 (0.9538) - - - * a21 -0.0059 -2.1433 (0.0324) - - - * a22 0.0861 2.6796 (0.0074) 0.4398 5.0872 (0.0000) * g11 0.9443 455.76 (0.0000) 0.9548 6996.4 (0.0000) * g12 -0.1111 -0.9989 (0.3184) - - - * g 21 0.0042 1.5418 (0.1232) - - - * g 22 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
  25. 25. Figure 1: News impact surface for the conditional stock index variance, full BEKK GARCH(1,1) model. 25
  26. 26. Figure 2: News impact surface for the conditional options strangle variance, full BEKK GARCH(1,1) model. 26
  27. 27. Figure 3: News impact surface for the conditional covariance, full BEKK GARCH(1,1) model. 27

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