نقش نوسانات ضمنی در پیش بینی صحیح نوسانات تحقق یافته و جهش در بازارهای ارز، سهام و اوراق قرضه

We study the forecasting of future realized volatility in the foreign exchange, stock, and bond markets from variables in our information set, including implied volatility backed out from option prices. Realized volatility is separated into its continuous and jump components, and the heterogeneous autoregressive (HAR) model is applied with implied volatility as an additional forecasting variable. A vector HAR (VecHAR) model for the resulting simultaneous system is introduced, controlling for possible endogeneity issues. We find that implied volatility contains incremental information about future volatility in all three markets, relative to past continuous and jump components, and it is an unbiased forecast in the foreign exchange and stock markets. Out-of-sample forecasting experiments confirm that implied volatility is important in forecasting future realized volatility components in all three markets. Perhaps surprisingly, the jump component is, to some extent, predictable, and options appear calibrated to incorporate information about future jumps in all three markets.

In both the theoretical and empirical finance literature, volatility is generally recognized as one of the most important determinants of risky asset values, such as exchange rates, stock and bond prices, and hence interest rates. Since any valuation procedure involves assessing the level and riskiness of future payoffs, it is particularly the forecasting of future volatility from variables in the current information set that is important for asset pricing, derivative pricing, hedging, and risk management. A number of different variables are potentially relevant for volatility forecasting. In the present paper, we include derivative prices and investigate whether implied volatilities (IV) backed out from options on foreign currency futures, stock index futures, or Treasury bond (T-bond) futures contain incremental information when assessed against volatility forecasts based on high-frequency (5-min) current and past returns on exchange rates, stock index futures, and T-bond futures, respectively. Andersen et al. (2003) and Andersen et al. (2004) show that simple reduced form time series models for realized volatility (RV) outperform commonly used GARCH and related stochastic volatility models in forecasting future volatility. In recent work, Barndorff-Nielsen and Shephard, 2004 and Barndorff-Nielsen and Shephard, 2006 derive a fully nonparametric separation of the continuous sample path (CC) and jump (JJ) components of RV. Applying this technique, Andersen et al. (2007) extend the results of Andersen et al. (2003) and Andersen et al. (2004) by using past CC and JJ as separate regressors when forecasting volatility. They show that the two components play very different roles in forecasting, and that significant gains in performance are achieved by separating them. While CC is strongly serially correlated, JJ is distinctly less persistent, and almost not forecastable, thus clearly indicating separate roles for CC and JJ in volatility forecasting. In this paper, we study high-frequency (5-min) returns to the $/DM exchange rate, S&P 500 futures, and 30 year T-bond futures, as well as monthly prices of associated futures options. Alternative volatility measures are computed from the two separate data segments, i.e., RV and its components from high-frequency returns and IV from option prices. IV is widely perceived as a natural forecast of integrated volatility over the remaining life of the option contract under risk-neutral pricing. It is also a relevant forecast in a stochastic volatility setting even if volatility risk is priced, and it should get a coefficient below (above) unity in forecasting regressions in the case of a negative (positive) volatility risk premium ( Bollerslev and Zhou, 2006). Since options expire at a monthly frequency, we consider the forecasting of one-month volatility measures. The issue is whether IV retains incremental information about future integrated volatility when assessed against realized measures (View the MathML sourceRV,C,J) from the previous month. The methodological contributions of the present paper are to use high-frequency data and recent statistical techniques for the realized measures, and to allow these to have different impacts at different frequencies, when constructing the return-based forecasts that IV is assessed against. These innovations ensure that IV is put to a harder test than in previous literature when comparing forecasting performance. The idea of allowing different impacts at different frequencies arises since realized measures covering the entire previous month very likely are not the only relevant yardsticks. Squared returns nearly one month past may not be as informative about future volatility as squared returns that are only one or a few days old. To address this issue, we apply the heterogeneous autoregressive (HAR) model proposed by Corsi (2009) for RV analysis and extended by Andersen et al. (2007) to include the separate CC and JJ components of total realized volatility (View the MathML sourceRV=C+J) as regressors. In the HAR framework, we include IV from option prices as an additional regressor, and also consider separate forecasting of both CC and JJ individually. As an additional contribution, we introduce a vector heterogeneous autoregressive (labeled VecHAR) model for joint modeling of View the MathML sourceIV,C, and JJ. Since IV is the new variable added in our study, compared to the RV literature, and since it may potentially be measured with error stemming from non-synchronicity between sampled option prices and corresponding futures prices, bid-ask spreads, model error, etc., we take special care in handling this variable. The simultaneous VecHAR analysis controls for possible endogeneity issues in the forecasting equations, and allows testing interesting cross-equation restrictions. Based on in-sample Mincer and Zarnowitz (1969) forecasting regressions, we show that IV contains incremental information relative to both CC and JJ when forecasting subsequent RV in all three markets. Furthermore, in the foreign exchange and stock markets, IV is an unbiased forecast. Indeed, it completely subsumes the information content of the daily, weekly, and monthly high-frequency realized measures in the foreign exchange market. Moreover, out-of-sample forecasting evidence suggests that IV should be used alone when forecasting monthly RV in all three markets. The mean absolute out-of-sample forecast error increases if any RV components are included in constructing the forecast. The results are remarkable considering that IV by construction should forecast volatility over the entire interval through expiration of the option, whereas our realized measures exclude the non-trading (exchange closing) intervals overnight and during weekends and holidays in the stock and bond markets. Indeed, the results most strongly favor the IV forecast in case of foreign currency exchange rates where there is round-the-clock trading. Squared returns over non-trading intervals could be included in RV for the other two markets, but with lower weight since they are more noisy. Leaving them out produces conservative results on the role of IV and is most natural given our focus on the separation of CC and JJ, which in practice requires high-frequency intra-day data. Using the HAR methodology for separate forecasting of CC and JJ, our results show that IV has predictive power for each. Forecasting monthly CC is very much like forecasting RV itself. The coefficient on IV is slightly smaller, but in-sample qualitative results on which variables to include are identical. The out-of-sample forecasting evidence suggests that IV again should be used alone in the foreign exchange and stock markets, but that it should be combined with realized measures in the bond market. Perhaps surprisingly, even the jump component is, to some extent, predictable, and IV contains incremental information about future jumps in all three markets. The results from the VecHAR model reinforce the conclusions. In particular, when forecasting CC in the foreign exchange market, IV completely subsumes the information content of all realized measures. Out-of-sample forecasting performance is about unchanged for JJ but improves for CC in all markets by using the VecHAR model, relative to comparable univariate specifications. The VecHAR system approach allows testing cross-equation restrictions, the results of which support the finding that IV is a forecast of total realized volatility View the MathML sourceRV=C+J, indeed an unbiased forecast in the foreign exchange and stock markets. In the previous literature, a number of authors have included IV in forecasting regressions, and most have found that it contains at least some incremental information, although there is mixed evidence on its unbiasedness and efficiency. 1 None of these studies has investigated whether the finding of incremental information in IV holds up when separating CC and JJ computed from high-frequency returns, or when including both daily, weekly, and monthly realized measures in HAR-type specifications. An interesting alternative to using individual option prices might have been to use model-free implied volatilities as in Jiang and Tian (2005). However, Andersen and Bondarenko (2007) find that these are dominated by the simpler Black–Scholes implied volatilities in terms of forecasting power. The remainder of the paper is laid out as follows. In the next section we briefly describe realized volatility and the nonparametric identification of its separate continuous sample path and jump components. In Section 3 we discuss the derivative pricing model. Section 4 describes our data. In Section 5 the empirical results are presented, and Section 6 concludes.

This paper examines the role of implied volatility in forecasting future realized volatility and jumps in the foreign exchange, stock, and bond markets. Realized volatility is separated into its continuous sample path and jump components, since Andersen et al. (2007) show that this leads to improved forecasting performance. We assess the incremental forecasting power of implied volatility relative to Andersen et al. (2007). On the methodological side, we apply the HAR model proposed by Corsi (2009) and applied by Andersen et al. (2007). We include implied volatility as an additional regressor, and also consider forecasting of the separate continuous and jump components of realized volatility. Furthermore, we introduce a vector HAR (VecHAR) model for simultaneous modeling of implied volatility and the separate components of realized volatility, controlling for possible endogeneity issues. On the substantive side, our empirical results using both in-sample Mincer and Zarnowitz (1969) regressions and out-of-sample forecasting show that in all three markets, option implied volatility contains incremental information about future return volatility relative to both the continuous and jump components of realized volatility. Indeed, implied volatility subsumes the information content of several realized measures in all three markets. In addition, implied volatility is an unbiased forecast of the sum of the continuous and jump components, i.e., of total realized volatility, in the foreign exchange and stock markets. The out-of-sample forecasting evidence confirms that implied volatility should be used in forecasting future realized volatility or the continuous component of this in all three markets. Finally, our results show that even the jump component of realized return volatility is, to some extent, predictable, and that option implied volatility enters significantly in the relevant jump forecasting equation for all three markets. Overall, our results are interesting and complement the burgeoning realized volatility literature. What we show is that implied volatility generally contains additional ex ante information on volatility and its continuous sample path and jump components beyond that in realized volatility and its components. This ex ante criterion is not everything that realized volatility may be used for, and it is possibly not the most important use. For example, realized volatility and its components can be used for ex post assessments of what volatility has been, whether there have been jumps in prices or not, etc. Implied volatility (even ex post implied volatility) is not well suited for these purposes.