Volume 7 Issue 9 - March 6, 2009
Clarifying the Dynamics of the Relationship between Option and Stock Markets Using the Threshold Vector Error Correction Model
Ming-Yuan Leon Li

Department of Accountancy, College of Management, National Cheng Kung University

Mathematics and Computers in Simulation, 2008, volume 79, issue 3, page 511–520

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There has been a significant body of research on the interrelation between option and stock markets, including Panton (1976, Journal of Econometrics), Anthony (1988, Journal of Finance), Bakshi et al. (2000, Review of Financial Studies), Manaster and Rendleman (1982, Journal of Finance), Bhattacharya (1987, Journal of Financial and Quantitative Analysis), Stephan and Whaley (1990, Journal of Finance) and Chan et al. (1993, Journal of Finance). Unlike previous studies, this investigation employs a new framework for examining questions regarding the dynamic relationships between option and stock markets.

The ideas of this study are presented as follows. Theoretically, the implied stock price calculated using the Black-Scholes [12] model (hereafter, the BS model) and the observed stock price should converge and arbitrage opportunities should not exist. However, owing to possible misspecifications of the BS model and market imperfections, this study establishes a nonlinear adjustment mechanism in which the speed of convergence toward the equilibrium relationship increases with absolute price discrepancies.

Building upon the above, this work investigates nonlinear relationships between option and stock markets using a regime-varying framework. Briefly, this study uses the threshold VECM (vector error correction model) to define two different market regimes based on magnitude of price deviation: (1) when the deviation between the implied and the observed stock prices exceeds a critical threshold the benefits of adjustment exceed the costs, and thus economic agents take significant actions to restore system equilibrium; (2) when the price discrepancy falls below the critical threshold, the convergence towards the equilibrium is unremarkable.

In particular, the threshold VECM used in this study is specified below:

, where △ denotes the difference operator (such as, △st=st-st-1), while st and s*t are the natural log prices of the OBS and IMP prices at time t, respectively. Notably, this investigation sets the EC term, zt-1 to (st-1-λ0-λ1s*t-1), which represents the last period disequilibrium between st-1 and s*t-1 prices and the θ denotes the threshold parameter.

Two market regimes are defined in the above setting: (1) regime I or the central regime (namely k=1), where |zt-1|≦θ, while (2) regime II or the outer regime (namely k=2), where |zt-1|>θ. The dynamics of the mispricing term, zt, depend on the market regime where that term is located. Particularly, in the central regime: –θ<zt-1<θ (namely, k=1), the mispricing term, zt-1 is too small to trigger arbitrage trading; however, in the outer regime: zt-1<–θ or zt-1>θ (namely, k=2), the mispricing term, zt-1 is sufficiently large to initiate arbitrage trading and thus the OBS-IMP deviation from the equilibrium condition, zt-1, helping to significantly affect the next period values of the IMP and OBS prices. Consequently, the estimates of βk=2 and β*,k=2 based on regime II are expected to be significant, while the estimates of βk=1 and β*,k=1 for regime I are expected to be insignificant.

Next, by considering two pairs of error terms, namely (ek=1t and e*,k=1t) and (ek=2,t and e*,k=2t), the two sets of covariance matrix are established as follows:
The threshold VECM has two key advantages compared to the conventional VECM. First, the discrete adjustments in the threshold system release the unrealistic assumption that the tendency of the option-stock market moving towards equilibrium exists during every time period. Second, the threshold VECM considers the concept of arbitrage threshold and applies it to identify the outer/central regime at each time point, and then calculates the variance and correlation parameters for each regime. The threshold VECM can thus overcome the limitations of constant variance and correlation.

Notably, this investigation is one of the first studies on the application of threshold systems to the dynamics of the interrelation between the option and stock markets. Moreover, compared to previous studies in the literature of application of threshold models, this study not only investigates the impacts of price transmission mechanisms on stock return means but also the volatilities of returns. The model is tested using the U.S. S&P 500 stock market. The sample period of the daily stock price is from 2002 to 2005. The empirical findings of this investigation are consistent with the following notions. First, the equilibrium re-establishment process depends primarily on the option market and is triggered only when price deviations exceed a critical threshold. Second, arbitrage behaviors between the option and stock markets increase volatility in these two markets and reduce their correlation.
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