واکنش نادرست یا مشخص سازی نادرست؟بررسی دوباره ناهنجاری نوسانات

Existing research examines the impact of volatility shocks on the relative pricing of long-term vs. short-term options and documents patterns of “short-horizon underreaction” and “long-horizon overreaction” in the options market. These studies, however, rely on implied volatilities derived from specific option-pricing models and are thus subject to model specification errors. In this paper, we show that these anomalous patterns are the result of model misspecification as opposed to market misreaction. We provide evidence that these patterns are consistent with, in both direction and magnitude, inherent biases in the misspecified models. We also apply a model-free approach to re-examine the anomalous patterns and find no evidence of market misreaction.

Option implied volatility reflects investors’ expectations of future volatility over the remaining life of the option. As new information arrives, investors revise their expectations of future volatility leading to changes in implied volatility. If investors formulate their expectations rationally, the level of and change in implied volatility should fully incorporate the set of information accumulated to date and newly arrived information. Previous empirical research (e.g., Stein’s, 1989 and Poteshman’s, 2001) examines the impact of volatility shocks on the relative pricing of long-term vs. short-term options and documents certain anomalous patterns. This is interpreted as market misreaction to volatility shocks. These studies, however, rely on implied volatilities derived from specific option-pricing models and are thus subject to model specification errors. In this paper, we argue that these anomalies are the result of model misspecification as opposed to market misreaction. We provide evidence that these patterns are consistent with both the direction and magnitude of pricing biases inherent in the specific models assumed by previous studies. We also implement a model-free approach to re-examine the anomalous patterns and find no evidence of market misreaction. Parallel to the literature on stock market anomalies, previous empirical studies document misreaction to volatility shocks in the options market. For example, Stein (1989) derives volatility term structure relations under the assumption of a mean-reverting, AR(1)-type stochastic volatility model and empirically tests the theoretical relationship using Black and Scholes (1973, B-S) implied volatilities from the S&P 100 (OEX) index options during the period from December 1983 to September 1987. He finds that the estimated slope coefficient of the volatility term structure is significantly higher than predicted and the forecast error for future volatility is negatively correlated with the short-term volatility. These results are interpreted as evidence that long-term volatility overreacts to changes in short-term volatility. In a subsequent study, Poteshman (2001) performs further tests on options market misreaction by examining implied volatilities from the S&P 500 (SPX) index options during the period from June 1988 to August 1997. In addition to using data from a different options market and time period, he also performs new tests of market misreaction by fitting Heston’s (1993) stochastic volatility model (the Heston model, hereafter) to option prices. He then extracts instantaneous volatilities from the fitted model using both full sample and subsets of short-term and long-term options. Comparing instantaneous volatilities extracted from short-term and long-term options, he examines how volatility shocks are projected over the long horizon relative to the short horizon. His findings support Stein’s (1989) overreaction hypothesis and a new pattern of market underreaction at short horizons. He concludes that these mixed patterns of “long-horizon overreaction” and “short-horizon underreaction” are consistent with investors who have established prior beliefs and are initially slow to recognize/react to volatility shocks and yet are prone to overreact once they do recognize the volatility shocks. 1 Evidence on volatility anomalies is mixed, however. The findings may vary depending on model specification and choice of option markets and sample periods. For example, Campa and Chang (1995) empirically examine volatility term structure under the Hull and White (1987) stochastic volatility model using daily currency option prices from December 1, 1989 to August 31, 1992. They find little evidence that long-term volatility overreacts to innovations in short-term volatility. Heynen et al. (1994) examine volatility term structure under three alternative model specifications – mean-reverting, GARCH, and EGARCH volatility models. Their mean-reverting model is essentially the same AR(1) model assumed by Stein (1989). They empirically test the derived term structure relations using equity and equity index options and find that the results vary substantially across model specifications. Although overreaction cannot be rejected under either the mean-reverting model or the GARCH model, it is rejected under the EGARCH model. In other words, model specification is critical and market misreaction may or may not be rejected depending on the assumed volatility model. The mixed findings on market misreaction is not entirely unexpected. All previous studies have relied on implied volatilities calculated using specific option-pricing models such as the B-S model or the Heston model. Their findings are based on joint tests of the expectations hypothesis and the assumed option-pricing model and are thus subject to model specification errors. Empirical studies such as Bates’s, 1996, Bates’s, 2000, Bates’s, 2006, Bakshi et al.’s, 1997, Bakshi et al.’s, 2000, Andersen et al.’s, 2002, Pan’s, 2002, Carr and Wu’s, 2003, Chernov et al.’s, 2003 and Eraker et al.’s, 2003 provide evidence of severe misspecification in the B-S model and commonly used stochastic volatility models. Although specification errors can potentially have significant effects on tests of market misreaction (e.g., Heynen et al., 1994), the existing literature has not addressed the issue of how model misspecification may lead to conflicting findings on market misreaction. In this paper, we examine the impact of misspecified models on tests of market misreaction and provide an explanation for the conflicting findings in previous studies. In particular, we focus on specification errors of the B-S model and the Heston model since most previous studies perform misreaction tests using implied volatilities based on these two models. We show that both models impose specific structures on the relationship between implied volatilities of long-term and short-term options. Such model-specific relations have direct implications for misreaction tests. We provide both theoretical arguments and Monte Carlo analysis to demonstrate that the patterns of long-horizon overreaction and short-horizon underreaction are consistent with the direction and magnitude of model specification errors inherent in these models. Even in the absence of any market misreaction, these model-specific relations may lead to the false identification of anomalous patterns in implied volatility. More importantly, we adopt a model-free approach and perform new tests of market misreaction. In contrast to previous studies, we do not make any specific assumptions about the underlying asset price process or make use of any option-pricing models. We thus avoid the problems arising from joint tests of the null hypothesis and the assumed model in previous studies. The key ingredient of our approach is the use of model-free implied volatility2 that is extracted directly from market prices of options across strike prices. In addition, the model-free implied volatility is also informationally more efficient than the B-S implied volatility (e.g., Jiang and Tian, 2005). Empirical tests using the model-free implied volatility are thus also more efficient than corresponding tests using the B-S implied volatility.3 Using SPX options data from June 1, 1988 to December 31, 2007, we perform model-free tests of market misreaction as well as their model-based counterparts. By performing both model-based and model-free versions of the same test, it is possible to disentangle the effect of model misspecification and misreaction. Replicating Stein’s model-based test, we find a similar negative relationship between the forecast error of future volatility and the short-term volatility. This is consistent with Stein’s finding of long-term volatility overreacting to changes in short-term volatility. In contrast, the model-free version of the same test provides no evidence of any negative (or positive) relationship, rejecting the overreaction hypothesis. Removing model specification problems associated with previous studies, all evidence of market misreaction disappears. Combining results from both model-free and model-based tests, it is clear that the model-based tests can erroneously reject the null hypothesis when model-free tests do not. This finding casts doubt on the validity of volatility anomalies identified in previous studies. When a model-based test finds evidence of negative correlation, it does not necessarily indicate market misreaction. Model-free tests should be performed as well to corroborate the findings. One must be cautious when drawing conclusions from the results of model-based tests. The rest of the paper is organized as follows. Section 2 provides a review of the relevant literature on options market misreaction and motivates the present study. In Section 3, we examine the impact of model specification errors on the misreaction tests in previous studies. Model-free tests of market misreaction are described and implemented in Section 4. The final section concludes.

Our analysis in this paper provides new insight on tests of volatility anomalies in the options market. As volatility shocks are not directly observable, past empirical research usually analyzes the differential effect of volatility shocks on long-term vs. short-term options. An option-pricing model is used to assess the expected impact of volatility shocks and how it differs for long-term vs. short-term options. Comparing the actual and expected effects, these studies identify volatility anomalies and draw conclusions on the informational efficiency of the options market. Such model-based tests are thus joint tests of market efficiency and the assumed option-pricing model. A rejection of the null hypothesis is not necessarily evidence against market efficiency due to model misspecification problems. In particular, we re-examine two volatility anomalies documented in previous studies (e.g., Stein’s, 1989 and Poteshman’s, 2001) that suggest “long-horizon overreaction” and “short-horizon underreaction” in the options market. These studies rely on implied volatilities derived from the B-S or Heston model, however, and are thus subject to model specification errors. We evaluate of the impact of model misspecification on these tests of market misreaction and assess both the direction and magnitude of the biases on test results. We show that these anomalous patterns are likely the result of model misspecification as opposed to market misreaction. We also perform model-free tests of market misreaction to volatility shocks and find no evidence against the informational efficiency of the options market. Our results suggest that one must be careful interpreting results from joint tests of the expectations hypothesis and specific option-pricing models. Every effort must be made to ensure that model misspecification is not responsible for rejecting the null hypothesis. Model-free tests of the null hypothesis should be attempted whenever possible.