Examining the interrelation dynamics between option and stock markets using the Markov-switching vector error correction model

Abstract

This study examines the dynamics of the interrelation between option and stock markets using the Markov-switching vector error correction model. Specifically, we calculate the implied stock prices from the Black–Scholes 6 model and establish a statistic framework in which the parameter of the price discrepancy between the observed and implied prices switches according to the phase of the volatility regime. The model is tested in the US S&P 500 stock market. The empirical findings of this work are consistent with the following notions. First, while option markets react more quickly to the newest stock–option disequilibrium shocks than spot markets, as found by earlier studies, we further indicate that the price adjustment process occurring in option markets is pronounced when the high variance condition is concerned, but less so during the stable period. Second, the degree of the co-movement between the observed and implied prices is significantly reduced during the high variance state. Last, the lagged price deviation between the observed and implied prices functions as an indicator of the variance-turning process.

Keywords:

• option market
• Markov-switching
• error correction model
• volatility

JEL Classification: :

• G12
• G13
• C51

Acknowledgements

The authors would like to make an acknowledgement to the two anonymous reviewers for their very helpful comments and gratefully acknowledge funding from the National Science Council of Taiwan (NSC96-2416-H-006-023-MY3).

Notes

Certain characteristics of option markets listed by such studies include: (1) options have lower trading costs; (2) unlike stocks, investors are not prohibited from shorting the call options (or buying the put options) when the underlying stock is on a down tick; and (3) by investing in options, investors can obtain higher leverage.

When OBS and IMP are cointegrated, the lagged OBS–IMP deviation should be included in the model, after which the VECM specification is generated. This lagged OBS–IMP deviation component is also known as the EC or mispricing error term.

The DataStream database provides data for the at-the-money options with different months to maturity, such as one, three and six months.

We adopt a basic form of the BS model without the consideration of the dividend payment. Merton 48 considers the effect of the dividend payment and modifies the BS formulas by incorporating the dividend yield variable. Collecting the dividend yields of all the S&P 500 companies and calculating the weighted dividend yield are very time-consuming. Further, the target of our work is not to find the best BS model. Therefore, for reasons of convenience, we adopt the BS formula exactly as it was created by Black and Scholes 6 to calculate the implied S&P 500 stock index.

The at-the-money option with a constant time to maturity of one month is adopted in this study. The corresponding T parameter is thus defined as 1/12.

The program codes of IMP calculation are edited by the MATLAB, version 6-5. The MATLAB program codes are available on demand.

We ever employed the SWARCH model proposed by Hamilton and Susmel 25 to calculate the variance. Nonetheless, our key conclusions regarding the price transmission process between the stock and option markets across various variance states are unchanged because the differences in the parameter estimates of the EC terms are statistically and economically marginal.

After controlling the time-varying variances using the GARCH(1,1) model, the skewness and kurtosis coefficients of the standardized error term (i.e. in Equation (5)) are −0.0385 (close to zero) and 3.1607 (close to three), respectively. Further, the Jarque–Bera (JB) statistic for the null hypothesis of normality assumption on the error term v _t is 1.4433 (p=0.4859). Consequently, the null hypothesis of normality on v _t in Equation (5) could not be rejected using the 1% significance level.

Phillips and Perron 50 propose nonparametric alternatives to the ADF test.

In the cointegration test, the intercept term without the time trend in the EC term is defined and the lag intervals in vector autoregression (VAR) are set as four. To save space, the further details of Johansen's 28 29 cointegration test are not presented; nonetheless, the results are available on demand.

For convenience, we assume p=q in Equations (7) and (8).

In accordance with the long-term adjustments to disequilibria, when z _t−1>0, then s _t should decrease and should increase to return the price relationship to the long-run equilibrium, while the opposite occurs when z _t−1<0. Therefore, the signs of z _t−1 in the OBS and IMP equations should be negative and positive, respectively.

In practice, option markets are assumed to respond more quickly to information than spot markets due to certain characteristics of the former.

We apply the JB tests on the residual terms of the VECM in which constant variances and correlations are defined. It must be noted that the JB statistics of the residual terms in the OBS and IMP returns (i.e. in Equations (7) and (8)) are 283.7936 (p=0.0000) and 1244.159 (p=0.0000), respectively, and thus the null hypothesis of normality is rejected at the 1% significance level. In other words, we indicate that the VECM in which constant variances and correlations are defined is not correctly specified.

It must be noted that although we ever extended the number of lagged terms in the VECM to four, the key conclusion is unchanged.

Because of the similarity to the VECM in Equations (7) and (8), we assume p=q in Equations (12) and (13).

In general, the OBS and IMP almost perfectly follow one another, since they represent the same asset; we thus establish the dual state specifications regarding the HV/LV states. We ever established a system with quarterly states in which both the OBS and IMP returns are subject to their own volatility state-switching processes. However, the HV–LV and LV–HV states are not well defined and are a collection of spikes.

We extend Hamilton and Susmel's 25 one-dimensional SWARCH model to establish a system with two dimensions.

The Markov-switching mechanism and the GARCH model are widely considered impossible to combine, with the exception of Gray 23, who establishes a system that permits such a combination on the basis of specific assumptions. See Hamilton and Susmel 25 for a more in-depth discussion of this issue.

If the OBS and IMP differ, arbitrage trading between the stock and option markets will be triggered regardless of whether the price deviation is positive or negative. Consequently, we use the absolute value of the spot–option price deviation as an indicator of market variances. Notably, this study ever used several different settings on the variable; for instance, the weighting of the most recent, second most recent and third most recent days follows the pattern 1:1:1. However, our key conclusions are unchanged, although the prediction performance of the lagged OBS–IMP deviations on the high variance is statistically marginal (i.e. the θ _p1 estimate is insignificant).

The logistic cumulative distribution, as in Equations (22) and (23), has been used almost exclusively as the TVTP functions (see e.g. 13 17 32 and many others). Exceptions only include Filardo and Gordon 18 and Kim and Yoo 33, where a probit function was adopted. The logistic TVTP function is appealing in that it is flexible and has a sensible economic interpretation, although one might encounter nonconvergence problems with the use of the logistic function.

It must be noted that the MSVECM uses the Markov-switching mechanism to control the structural changes in return volatility during the test period, and hence effectively mitigate the non-normality problems of the return rates. In particular, after controlling the regime-switching process in return variance, the JB statistics of the standardized residual terms in the OBS and IMP returns (i.e. and in Equations (12) and (13)) are 0.716 (p=0.699) and 0.874 (p=0.721), respectively. This result indicates that the null hypothesis of normality could not be rejected.

More specifically, when the information set used for estimation includes signals dated up to time t, the regime probability is a filtered probability. It is also possible to use the overall sample period information set to estimate the state probability at time t. The probability is denoted as a smoothed probability. In contrast, a predicting probability denotes the regime probability for an ex ante estimation, with the information set including signals dated up to period t−1.

Even with this simple structure involving two lagged ARCH components, there are 24 parameters requiring estimation. A more general structure with higher-order ARCH terms could increase the number of parameters to be estimated. This study also found that the higher-order ARCH parameter estimates in the SWARCH model generally do not differ significantly from zero. To save space, we do not report the results of the higher lag order setting.

The algorithm of Boyden, Fletcher, Goldfarb, and Shanno (BFGS) can effectively yield the maximum value of the nonlinear-likelihood functions, as demonstrated by Luenberger 41. Additionally, we randomly generate 50 sets of initial values and derive the ML function value for each of them. The mapped converged measure of the greatest ML function value is then used for the parameter estimation.

With the null hypothesis of one-regime setting against the alternative of two-regime setting, the conventional LR statistic under this condition no longer follows the standard χ2 distribution. Hansen 26 27 proposes a bound test that addresses these problems, but its computational difficulty has limited its applicability. We ever applied the Hansen's test to the OBS and IMP returns and found that the nonlinear dynamics of the OBS and IMP returns are dominated by variance-switching process rather than switching in means. The related test results are available on demand.

In particular, the and estimates in the OBS equation are 0.0564 (t-) and 0.0306 ($t$-statistics$ = $0.2757), respectively.

The high variance regimes of the 2001–2003 periods reflect the impacts of the 11 September 2001 terrorist attack on America. However, it must be noted that the duration of certain HV regimes is just a single day, particularly when the post-2003 period is concerned. This phenomenon might be due to stock prices adjusting instantaneously to new information and the TVTP used in this study.

Please refer to Li 37 for the related discussions.

In specific, the θ _q1 and θ _p1 estimates are −2.0455 (t-statistics = 4.6937) and 1.3594 (t-statistics = 2.940), respectively. It must be noted that there are no nonconvergence problems when we use the logistic TVTP function.

As is well known, a key feature of the nonlinear volatility-switching model is its ability to capture the rare extremes that occur in the OBS–IMP relationship, and that strongly influence the dynamics of the relationship between stock and option markets. Emerging stock markets generally experience more extreme crisis events than mature stock markets. Consequently, a comparative analysis of mature against emerging stock markets would be meaningful.