آیا آینده قیمت VIX قابل پیش بینی است؟ تحقیقات تجربی

This paper investigates whether volatility futures prices per se can be forecasted by studying the fast-growing VIX futures market. To this end, alternative model specifications are employed. Point and interval out-of-sample forecasts are constructed and evaluated under various statistical metrics. Next, the economic significance of the forecasts obtained is also assessed by performing trading strategies. Only weak evidence of statistically predictable patterns in the evolution of volatility futures prices is found. No trading strategy yields economically significant profits. Hence, the hypothesis that the VIX volatility futures market is informationally efficient cannot be rejected.

Volatility derivatives have attracted a considerable amount of attention in past years, since they enable trading and hedging against changes in volatility. Brenner and Galai, 1989 and Brenner and Galai, 1993 first suggested derivatives written on some measure of volatility that would serve as the underlying asset. Since then, a number of volatility derivatives have been traded in the over-the-counter market. On March 26, 2004, volatility futures on the implied volatility index VIX were introduced by the Chicago Board Options Exchange (CBOE).1 Volatility futures on a number of other implied volatility indices have also been introduced since then. The liquidity of volatility futures markets is steadily growing, with the VIX futures market being the most liquid one.2 This paper focuses on the VIX futures market and for the first time addresses the question of whether VIX futures prices per se can be predicted.3 The answer to the question of whether or not volatility futures prices can be predicted is of importance to both academics and practitioners, because it contributes to our understanding of whether volatility futures markets are efficient, and helps market participants to develop profitable volatility trading strategies and set successful hedging schemes. There is already an extensive body of literature that has investigated whether the prices of stock indexes, interest rates, currencies, and commodity futures can be forecasted. The significance of the results has been evaluated using either a statistical or an economic (trading profits) metric. A number of studies have documented a statistically predictable pattern in futures returns. In particular, Bessembinder and Chan (1992) found that the monthly nearest maturity commodity and currency futures returns can be forecasted within-sample in a statistical sense. They concluded that this predictability could be attributed to an asset pricing model with time-varying risk-premia. Similar findings were documented by Miffre (2001a) for the FTSE 100 futures and by Miffre (2001b) for commodity and financial futures. On the other hand, the empirical evidence on the predictability in futures markets under an economic metric is mixed. For instance, Hartzmark (1987) found that in aggregate, speculators do not earn significant profits in commodity and interest rate futures markets; daily data of all contract maturities were employed. Yoo and Maddala (1991), however, studied commodity and currency futures and found that speculators tend to be profitable; daily data for a number of futures maturities were considered. Similar findings were reported by Kearns and Manners (2004), Kho (1996), Taylor (1992) and Wang (2004). In particular, all of these studies found that economically significant profits can be obtained by employing various trading rules in currency futures markets; daily data were used by Taylor (1992), and weekly data by Kearns and Manners (2004), Kho (1996), and Wang (2004). A number of futures maturities were examined by Kearns and Manners (2004) and Taylor (1992), while Kho (1996) and Wang (2004) focused on the shortest maturity series. Significant profits were also reported by Hartzmark (1991) and Miffre (2002), who examined the commodity and financial futures markets; the latter study focused only on the shortest maturity contracts. Regarding the source of the identified trading profits, Kearns and Manners (2004) and Taylor (1992) attributed them to the inefficiency of the currency futures market. On the other hand, Kho (1996), Miffre (2002), Wang (2004) and Yoo and Maddala (1991) found that the reported profits were not abnormal, and Hartzmark (1991) found that profitability is determined by luck rather than superior forecast ability; hence, the considered markets were efficient á la Jensen (1978). In contrast to the number of papers devoted to the topic of predictability in the previously mentioned futures markets, the research as to whether there exist predictable patterns in the evolution of volatility futures prices is still at its infancy. The literature on volatility futures has primarily focused on developing pricing models (see e.g. Brenner et al., 2008, Dotsis et al., 2007, Grünbichler and Longstaff, 1996, Lin, 2007 and Zhang and Zhu, 2006) and assessing their hedging performances (see e.g. Jiang & Oomen, 2001). On the other hand, to the best of our knowledge, Konstantinidi, Skiadopoulos, and Tzagkaraki (2008) is the only related study that has explored the issue of the predictability of volatility futures prices. However, this was done indirectly, and only under a financial measure. The authors developed trading strategies with VIX and VXD volatility futures based on point and interval forecasts which were formed for the corresponding underlying implied volatility indices. They found that the Sharpe ratios obtained were not statistically different from zero, and hence the volatility futures markets are efficient. This study extends the literature on whether the evolution of volatility futures prices can be forecasted. In contrast to Konstantinidi et al. (2008), we investigate the predictability of the VIX volatility futures prices per se, without searching for predictable patterns in the underlying implied volatility index. This is because predictability in the underlying implied volatility index market does not necessarily imply that volatility futures prices can be predicted, since there may be other factors/information flows that affect volatility futures markets as well. This is analogous to the interest rate derivatives literature, where it is well documented that models which describe the dynamics of the underlying interest rate quite well, cannot account for the properties of the prices of the corresponding interest rate derivative (the “unspanned stochastic volatility problem”; see, e.g., Jarrow, Li, & Zhao, 2007, and references therein). In our case, the relationship between changes in the prices of VIX futures and its underlying index is not known a priori from a theoretical point of view; there is no cost-of-carry relationship in the case of VIX futures, since the underlying index is not a tradable asset. In addition, volatility futures prices may not always be moving in the same direction as the underlying implied volatility index, due to market microstructure effects (see a discussion and similar findings by Bakshi, Cao, & Chen, 2000, who conducted an analysis for call options using intra-day data). To address our research question, both point and bootstrapped interval out-of-sample forecasts are considered. This is because interval forecasts have been found to be useful for volatility trading purposes; for example, Poon and Pope (2000) found that profitable volatility spread trades can be developed in the S&P 100 and S&P 500 index option markets by constructing certain intervals. Using a number of tests and criteria, we test the statistical significance of the forecasts obtained. In addition, their economic significance is investigated by means of trading strategies. This is the ultimate test for concluding whether or not the recently inaugurated volatility futures market is efficient. To check the robustness of our results, the analysis is performed across various maturity futures series and by employing a number of alternative model specifications. The latter is necessary because the question of predictability is inevitably tested jointly with the assumed forecasting model. The remainder of this paper is structured as follows. Section 2 describes the data set, and Section 3 presents the forecasting models to be used. Section 4 discusses the results concerning the in-sample performances of the models under consideration. Next, the out-of-sample predictive performances of the various models are evaluated in statistical and economic terms in Sections 5 and 6, respectively. The last section concludes.

This paper has investigated, for the first time, whether volatility futures prices per se can be forecasted. To this end, the most liquid volatility futures market (futures on VIX) has been considered. A number of alternative model specifications have been employed: the economic variables model, the AR(2) model, the VAR model and the ARMA(1,1) model. Equally weighted and weighted combination forecasts have also been considered. Both point and bootstrapped interval forecasts have been constructed and their statistical and economic significance have been evaluated. The latter is assessed by means of trading strategies using the VIX futures. This has implications for the efficiency of the VIX volatility futures market. Regarding the statistical significance of the forecasts obtained, in the case of point forecasts, we found weak evidence of a statistically predictable pattern in the evolution of the shortest futures series. In the case of the interval forecasts, no model specification had predictive power. Regarding the economic significance of the forecasts obtained, the constructed forecasts did not yield economically significant profits. Overall, our results imply that one cannot reject the hypothesis that the VIX volatility futures market is informationally efficient. These findings are consistent with those of Konstantinidi et al. (2008), who studied the efficiency of the VIX futures market indirectly. On the other hand, our results are in contrast to those found about the efficiency of other futures markets (stock, currency, interest rate and commodities), where predictability in either statistical or economic terms has been documented. However, the fact that the VIX futures market is found to be efficient does not invalidate the trading of VIX futures. This is because VIX futures can also be used for hedging against changes in volatility. After all, this was the main motivation for their introduction (see Brenner and Galai, 1989 and Brenner and Galai, 1993). Future research should investigate the issue of predictability in volatility futures markets at longer horizons. It has been well documented that the predictability in asset returns increases as the horizon increases (see, e.g. Poterba & Summers, 1988). However, a longer horizon study is beyond the scope of this paper due to data limitations, as the VIX market has only been operating since 2004. Intra-day data should also be used to test whether any predictable patterns can be detected within the data; this will be particularly useful for day-traders. Finally, it may be worth considering more complex model specifications, given that the answer on the predictability question always depends on the assumed specification of the predictive regression.