would be expected to approach a normal or a stable **distribution** by central and generalized central limit theorem respectively, depending on whether the variance of the daily PDF exists or not. Single-day returns seem to be better described by power-law-tailed distributions (Fuentes, Gerig & Vicente, 2009; Gerig, Vicente & Fuentes, 2009; Ma & Serota, 2014) with existing variance, while intra-day **data** seem to point to very long tails with a diverging variance (Behfar, 2016) (with a usual caveat that the tail behavior is hard to pinpoint, especially with smaller **data** sets; for multi-day returns, see (Dashti Moghaddam & Serota, 2018; Liu et al., 2019)). Our own work (Dashti Moghaddam, Liu & Serota, 2019a) indicates that correlations fall off quickly, as a power law, over a period of about five days and then persist to slowly decay exponentially. Fig. 1 indicates a tailed **distribution** for RV 2 which saturates to its final shape over about five days as well. As per our current results (Dashti Moghaddam et al., 2019a), it is best fitted -- and with high precision -- by Generalized Beta Prime **distribution** – a generalization of BP – and Beta Prime **distribution**. Conversely, while VIX 2 and VXO 2 are best fitted by these two distributions as well, the precision is considerably worse, which may be another indicator of their deficiencies.

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To keep our discussion concise we have left out some important issues that can be explored in future work. We select two examples. First, in practice advances in **realized** **volatility** model- ing may not be translated so neatly into improvements in modeling the conditional **distribution** of returns. Two aspects of the link between **realized** **volatility** and returns should be studied more carefully. The assumption that returns standardized by **realized** **volatility** are approximately normal and independent seems to be inadequate for some series. Is there a role for jumps in ad- justing the **distribution**? Do the problems in measuring **realized** **volatility** make this relation less straightforward? We have also only considered a simple model for the dependence between return and **volatility** innovations. Second, we have mostly analyzed the performance of diﬀerent models in one day ahead applications. Because financial quantities are so persistent many incongruent models are misleadingly competitive at very short horizons. More emphasis should be placed in investigating whether diﬀerent models are consistent with a realistic longer horizon dynamics. Our analysis suggest that to do so we may need a more solid understanding of asymmetric eﬀects.

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Despite the sizable forecasting gains made possible by **volatility** models based on high frequency **data**, our descriptive results can be directly related to the failure of GARCH **volatility** models to completely account for the excess kurtosis of returns (see for example Malmsten and Ter¨asvirta, 2004, Carnero et al., 2004). The researcher or practitioner interested in evaluating the density of returns from the perspective of a time series model still lives in a fat tailed world and purely predictive models of **volatility** may have little to say about it. In this paper, we do not interpret those facts as evidence against those models, but as a consequence of two factors: **volatility** risk (which causes excess kurtosis in the ex ante **distribution** of returns) and **volatility** feedback (or intraday leverage) effects (causing negative skewness). In the following section, we will argue that an adequate **volatility** model for return density forecasting and risk management in this setting should illuminate the dynamics of the higher moments. To pursue this objective we will turn to the idea of studying the time series **volatility** of **realized** **volatility** following Corsi et al. (2008).

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We will present several arguments for this emphasis. First, the presence of high and time-varying **volatility** risk brings substantially more uncertainty to the tails of the **distribution** of asset returns. In a standard stochastic **volatility** setting where returns given **volatility** follow a gaussian **distribution** the degree of **volatility** risk is the main determinant of the size of the tails of the ex-ante **distribution** of returns. If future **realized** **volatility** is relatively unpredictable a focus on predictive models will be insufficient for obtaining a good grasp of the tails of the return **distribution**, which in many cases (e.g., in risk management applications) is the main objective of the econometrician. Intuitively, when there is substantial **volatility** risk the ex-post **realized** **volatility** will frequently turn out to be much higher than the forecasted values: tail returns that would be virtually impossible with the **distribution** based on the point forecast (sometimes used implicitly or explicitly as an approximation) may be observed. Even though returns standardized by ex-post quadratic variation measures are nearly gaussian, returns standardized by fitted or predicted values of time series **volatility** models are far from normal. Given the uncertainty in **volatility** this is expected and should not be seen as evidence against those models; explicitly modeling the higher moment is necessary.

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Tables 1-3 summarize the parameter estimation for the **volatility** risk premium. The use of model-free **implied** **volatility** (MFIV) achieves a similar root-mean-squared error (RMSE) and convergence rate as the true infeasible risk-neutral **implied** **volatility** (RNIV). On the other hand, the misspecified Black-Scholes **implied** **volatility** (BSIV) shows slow convergence in estimating the **volatility** risk premium. Also, using **realized** **volatility** from five-minute returns (over a monthly horizon) has virtually the same small bias and high efficiency as the estimates based on the (infeasible) integrated **volatility**. In contrast, using the **realized** **volatility** from daily returns generally results in larger bias and significantly lower efficiency. Figures 1-3 report the Wald test for the risk premium parameter, which should be asymp- totically X 2 (1) distributed. In the cases of (infeasible) integrated **volatility** and five-minute **realized** **volatility**, the test statistics for the MFIV and RNIV measures are generally indis- tinguishable and closely approximated by the asymptotic **distribution**, the only exception being the high **volatility** persistence scenario (b) for which the MFIV measure results in slight over-rejection. In contrast, the (misspecified) BSIV measure shows clear evidence of over-rejection for all of the different scenarios. When the **realized** **volatility** is constructed from daily squared returns, the Wald test systematically loses power to detect any misspeci- fication, and the RNIV and MFIV measures now both show some under-rejection bias, while the over-rejection bias for the BSIV measure is somewhat mitigated. 9

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Table 1A reports descriptive statistics for **volatility** series. Average values of returns and volatilities are compatible between all Indian stock indices. Skewness values are around 8 and kurtosis values at around 70. All statistic values are extreme; indicating lack of normality. So, returns are far from being normal; as expected. Fat tails are also evident. The kurtosis of returns is much higher than that of a normal **distribution** at intraday frequency and tends to decrease as the return length increases. Thus, the probability density functions (pdf) of returns are leptokurtic with shapes depending on the time scale and presenting a very slow convergence of the Central Limit Theorem to the normal **distribution**. These results are consistent with Jarque-Bera (JB) test; in which, normality is rejected in all series of returns. However, normality is not rejected by the Cramer-von Mises (CVM) test on returns. Moreover, the skewness and kurtosis values for volatilities are close to 3 and between 11 and 13, respectively; indicating distributions not strongly adverse to normality. However, normality is rejected by the CVM test and JB test on volatilities of all Indian stock indices.

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To keep our discussion concise we have left out some important issues that can be explored in future work. We select two examples. First, in practice advances in **realized** **volatility** model- ing may not be translated so neatly into improvements in modeling the conditional **distribution** of returns. Two aspects of the link between **realized** **volatility** and returns should be studied more carefully. The assumption that returns standardized by **realized** **volatility** are approximately normal and independent seems to be inadequate for some series. Is there a role for jumps in ad- justing the **distribution**? Do the problems in measuring **realized** **volatility** make this relation less straightforward? We have also only considered a simple model for the dependence between return and **volatility** innovations. Second, we have mostly analyzed the performance of diﬀerent models in one day ahead applications. Because financial quantities are so persistent many incongruent models are misleadingly competitive at very short horizons. More emphasis should be placed in investigating whether diﬀerent models are consistent with a realistic longer horizon dynamics. Our analysis suggest that to do so we may need a more solid understanding of asymmetric eﬀects.

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In order to assess the statistical signiÞcance of the regression estimates, we rely on subsam- pling (see Politis, Romano, and Wolf (1999) for a complete discussion). We prefer subsampling over the usual bootstrap because of its wider applicability. The only requirements for its valid- ity are the existence of a limiting **distribution** and some (rather mild) conditions limiting the dependence of either the **data** or the subsampled statistics. For example, subsampling is ap- plicable to the case of an autoregression with a unit root, while the standard bootstrap is not. Another advantage of subsampling over the bootstrap is that the rate of convergence to the asymptotic **distribution** does not have to be known and can be estimated (see Bertail, Politis, and Romano (1999)). This property is particularly attractive given that there is uncertainty as to whether we are in the stationary range or not and the convergence rates depend on the long memory parameters of both the regressors and the errors. Moreover, it is likely that the rates of convergence of the constant and slope estimators are diﬀerent. Our subsampling approach can estimate these diﬀerent rates consistently.

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Our results reveal that Canadian stock **market** experiences signi�cant price discontinues (jumps) and average jump arrival rate is about 0.17 jumps per day. We �nd that about 55% of jumps are due to the overnight returns and about 90% of jumps occur within 30 minutes of the **market** opening for trading – providing a strong evidence of jump clustering. While looking at the jump intensi- ties, our results show an asymmetric **distribution** of positive versus negative jumps for intraday returns but such asymmetry disappears when we include overnight returns in our analysis. Berk- man et al. (2012) and Lou et al. (2018) suggest that institutional investors tend to trade relatively more during the day and individual investors trade relatively more overnight. Such di�erences in jump charecteristics in intraday versus overnight returns potentially re�ecting the corresponding clientele e�ects. Therefore, it is important to incorporate overnight returns in jump risk analysis. 1 In our paper, we further show that although the e�ect of jump component in **volatility** fore- casting is statistically signi�cant, its economic signi�cance is very nominal - large portion of re- alized **volatility** is coming from the continuous component. When we examine e�ect of **market**

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we need the best possible predictor for the future return covariance matrix. Here, we have different choices for the estimation of the return **volatility** vector, as well as dif- ferent estimators for the correlation matrix. Roughly speaking, **volatility** estimators can be grouped into estimators based on historical time-series observations or **implied** **volatility** estimators. Reliance on risk-neutral estimates requires careful consideration of several points. At least two opposing effects come into play. On the one hand, im- plied estimators are based on empirically observable **market** prices. Therefore, they may be considered purely forward-looking variables with a high ability to reflect changing **market** conditions in a timely way. Thus, **implied** return moments are less prone to the statistical inertia of sample return time series. On the other hand, estima- tors under the risk-neutral measure reflect investor sentiment at the time of portfolio construction. As such, they can substantially differ from the **realized** values in the future. It is well-known that **implied** **volatility** typically overestimates future realiza- tions. This phenomenon entered the financial literature as **volatility** risk. 3 In addition, liquidity effects contained in option prices may distort **implied** moment estimators. To investigate the relative importance of the two opposing effects, we implement esti- mators on the historical, as well as the **implied** probability measures. Summarizing, it is not clear ex ante whether estimators relying on **implied** **volatility** outperform those relying on the sample time series. 4

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Christensen and Prabhala (1998) set out to check out the results presented by Canina et al. (1993). Time period for the S&P 100 index options price **data** used in the study over- laps with Canina et al. (1993) as the **data** is for 139 months period from November 1983 to May 1995. Options included in the **data** are at-the-money call options. They found that the **implied** **volatility** predicts the future **volatility** better than the **realized** **volatility** when forecasting. There are three reasons given for the different results when compared to the results presented by Canina et al. (1993). The first reason is that Christensen et al. have longer time period for the **data** to use in the study. The second reason is the **data** sampling as authors use a monthly **data** for the option and index prices, and options are those that are expiring just before the next sample date. The sampling made in this way gives the results some robustness against the autocorrelation in a daily returns. The third reason is that according to the authors the October 1987 stock **market** crash caused a shift in both **implied** and **realized** **volatility** levels. After the crash the explanatory value of the **implied** **volatility** for the future **volatility** is significantly better than before the crash. The result of Christensen et al. is supported by Gwilym and Buckle (1999) on one-month-forward forecasts.

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Filis (2009) analyzed the relationship between **implied** **volatility** and **realized** **volatility** in the Greek derivatives **market** using call and put option **implied** **volatility**. Daily **data** from January 2000 to January 2003, obtained from the Athens Derivatives Exchange and Athens Stock Exchange, were used to calculate the at-the-money call and put option **implied** volatilities. The results indicated that the **implied** **volatility** was a biased and inefficient predictor for the **realized** **volatility**, reflecting that the Greek derivatives **market** was inefficient. A unique study in relation to the informational content of **implied** **volatility** was carried out by Viteva et al. (2014) using the carbon futures traded on the European Climate Exchange (ECX), for the period from January 2008 to December 2010. The ECX is the world’s largest carbon derivatives exchange, on which carbon dioxide emissions are traded between companies that have installed climate-friendly technology and reduced their carbon emissions by more than is required and companies that exceed their emission limits. They found **implied** **volatility** to be highly informative about future **volatility** but a biased forecast of future **volatility** over the remaining life of the option. However, the forecast provided by the **implied** **volatility** was found to be statistically significant in predicting future **volatility** changes. Another recent and distinct study came from Birkelund et al. (2015), who developed an **implied** **volatility** index for the Nordic power forwards **market**, an electricity-linked **implied** **volatility** index. They used high-frequency **data** (every 30 minutes) to develop their daily observations, with a sample period from October 2005 to September 2011. The results suggested that there was a risk premium in the option contract price and that the **volatility** index was a biased estimator of the future **realized** **volatility**. The authors claimed that their findings were similar to those of Ederington and Guan (2002b).

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The analysis of nancial **market** **volatility** is of the utmost importance to asset pricing, derivative pricing, hedging, and risk management. Several di erent sources of information may be invoked in generating forecasts of unknown future **volatility**. Besides measurements based on historical return records, observed derivative prices are known from nance theory to be highly sensitive to and hence informative about future **volatility**. It is therefore natural to consider **data** on both asset prices and associated derivatives when measuring, modelling and forecasting **volatility**. Earlier literature has shown that **implied** **volatility** backed out from option prices provides a better **volatility** forecast than sample **volatility** based on past daily returns, but more recent literature shows that **volatility** forecasting based on past returns may be improved dramatically by using high-frequency (e.g., 5-minute) returns, and explicitly allowing for jumps in asset prices when computing forecasts. The important question addressed in the present paper is whether **implied** **volatility** from option prices continues to be the dominant **volatility** forecast, even when comparing to these new improved return based alternatives, using high-frequency **data** and accommodating a jump component in asset prices.

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C is for bond **data** and the results in the …rst line show that only monthly RV is signi…cant. In row two of each panel of Table 1, x = (C; J ), so this is a monthly frequency HAR- RV-CJ model (Andersen et al. (2007)). The conclusions for C are similar to those for RV in the …rst row, except that the monthly and weekly components become insigni…cant in the stock **market**. The jump components are insigni…cant, except daily J in the stock and bond markets. Adjusted R 2 improves when moving from …rst to second line of each panel of Ta- ble 1, thus con…rming the enhanced in-sample (Mincer-Zarnowitz) forecasting performance obtained by splitting RV into its separate components also found by Andersen et al. (2007). Out-of-sample forecasting performance (MAFE) improves in the stock **market** when sepa- rating C and J , but remains unchanged in the bond **market**, and actually deteriorates in the currency **market**, hence showing the relevance of including this criterion in the analysis. Next, **implied** **volatility** is added to the information set at time t in the HAR regressions. When RV is included together with IV , fourth row, all the **realized** **volatility** coe¢ cients turn insigni…cant in the foreign exchange and bond markets, and only daily RV remains sig- ni…cant in the stock **market**. Indeed, IV gets t-statistics of 6:15, 6:84, and 4:46 in the three markets, providing clear evidence of the relevance of IV in forecasting future **volatility**. The last row of each panel shows the results when including C and J together with IV , i.e., the full HAR-RV-CJIV model (14). In the foreign exchange **market**, IV completely subsumes the information content of both C and J at all frequencies. Adjusted R 2 is about equally high in the third line of the panel, where IV is the sole regressor and where also MAFE takes the best (lowest) value in the panel. In the stock **market**, both daily components of RV remain signi…cant, and the adjusted R 2 increases from 62% to 68% relative to having IV as the sole regressor, but again MAFE points to the speci…cation with only IV included as the best forecast. In the bond **market**, IV gets the highest t-statistic, as in the other two markets. In this case, the monthly jump component J t 22;t is also signi…cant and adjusted

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Other empirical studies on the link between returns and **volatility** are based on lower-frequency **data** or model-based measures of **volatility**; see Christie (1982), French et al. (1987), Schwert (1989), Turner, Startz and Nelson (1989), Nelson (1991), Glosten, Jagannathan and Runkle (1993) and Campbell and Hentschel (1992), Bekaert and Wu (2000), Whaley (2000), Ghysels, Santa-Clara and Valkanov (2004), Giot (2005), Ludvigson and Ng (2005), Dennis, Mayhew and Stivers (2006), and Guo and Savickas (2006) among others. On the relationship and the relative importance of the leverage and **volatility** feedback effects, the results of this literature are often ambiguous, if not contradictory. In particular, studies focusing on the leverage hypothesis conclude that the latter cannot completely account for changes in **volatility**; see Christie (1982) and Schwert (1989). However, for the **volatility** feedback effect, empirical findings conflict. French et al. (1987), Campbell and Hentschel (1992) and Ghysels et al. (2004) find a positive relation between **volatility** and expected returns, while Turner et al. (1989), Glosten et al. (1993) and Nelson (1991) find a negative relation. From individual-firm **data**, Bekaert and Wu (2000) conclude that the **volatility** feedback effect dominates the leverage effect empirically. The coefficient linking **volatility** to returns is often not statisti- cally significant. Ludvigson and Ng (2005) find a strong positive contemporaneous relation between the conditional mean and conditional **volatility** and a strong negative lag-**volatility**-in-mean effect. Guo and Savickas (2006) conclude that the stock **market** risk-return relation is positive, as stipulated by the CAPM; however, idiosyncratic **volatility** is negatively related to future stock **market** returns. Giot (2005) and Dennis et al. (2006) use lower frequency **data** (such as, daily **data**) to study the relationship between returns and **implied** **volatility**. Giot (2005) uses the S&P 100 index and an **implied** **volatility** index (VIX) to show that there is a contemporaneous asymmetric relationship between S&P 100 index returns and VIX: negative S&P 100 index returns yield bigger changes in VIX than do positive returns [see Whaley (2000)]. Dennis et al. (2006), using daily stock returns and innovations in option-derived **implied** volatilities, show that the rela- tion between stock returns and innovations in systematic **volatility** (idiosyncratic **volatility**) is substantially negative (near zero).

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variables. [10] evaluated the role of the online search activity for forecasting **realized** **volatility** of financial markets and commodity markets using models that also include **market**-based variables. They found that Google search **data** play a minor role in pre- dicting the **realized** **volatility** once **implied** **volatility** is included in the set of regressors. Therefore, they suggested that there might exist a common component between **implied** **volatility** and Internet search activity: in this regard, they found that most of the predictive information about **realized** **volatility** contained in Google Trends **data** is also included in **implied** **volatility**, whereas **implied** **volatility** has additional predictive content that is not captured by Google **data**.

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a stochastic **volatility** (SV) model estimated with **data** at daily or lower fre- quency. An alternative approach is to invoke option pricing models to invert observed derivatives prices into **market**-based forecasts of “**implied** **volatility**” over a fixed future horizon. Such procedures remain model-dependent and further incorporate a potentially time-varying **volatility** risk premium in the measure so they generally do not provide unbiased forecasts of the **volatility** of the underlying asset. Finally, some studies rely on “historical” **volatility** measures that employ a backward looking rolling sample return standard de- viation, typically computed using one to six months of daily returns, as a proxy for the current and future **volatility** level. Since **volatility** is persistent such measures do provide information but **volatility** is also clearly mean re- verting, implying that such unit root type forecasts of future **volatility** are far from optimal and, in fact, conditionally biased given the history of the past returns. In sum, while actual returns may be measured with minimal (measurement) error and may be analyzed directly via standard time series methods, **volatility** modeling has traditionally relied on more complex econo- metric procedures in order to accommodate the inherent latent character of **volatility**.

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In Figure 2, we plot the time series of 1Y at-the-money IV of DAX index options (left axis, black line) together with DAX closing prices (right axis, gray line). An option is called at-the-money (ATM) when the exercise price is equal or close to the spot (or to the forward). The index options were traded at the EUREX, Frankfurt (Germany), from 2000 to 2008. As is visible IV is subject to considerable variations. Average DAX index IV was about 22% with signiﬁcantly higher levels in times of **market** stress, such as after the World Trade Center attacks 2001, during the pronounced bear **market** 2002-2003 and the ﬁnancial crisis end of 2008.

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We use daily **data** from the 1 st of February, 2001 up to the 9 th of July, 2013 (i.e. 3132 trading days) from eight **implied** **volatility** indices. The **implied** volatilities are the following: VIX (S&P500 **Volatility** Index – US), VXN (Nasdaq-100 **Volatility** Index – US), VXD (Dow Jones **Volatility** Index – US), VSTOXX (Euro Stoxx 50 **Volatility** Index – Europe), VFTSE (FTSE 100 **Volatility** Index – UK), VDAX (DAX 30 **Volatility** Index – Germany), VCAC (CAC 40 **Volatility** Index – France) and VXJ (Japanese **Volatility** Index - Japan). The stock markets under consideration represent six out of the ten most important stock markets internationally, in terms of capitalization. In addition, these markets are among the most liquid markets of the world. Thus, we maintain that their **implied** **volatility** indices are representative of the world’s stock m arket uncertainty. The **data** were extracted from Datastream ® . As we aim for a common sample of the aforementioned **implied** **volatility** indices, the starting **data** of the sample period were dictated by the availability of the **data** of the VXN index.

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list. First, we have the indicator section which keeps the indicators used to predict the stocks by (default is only VIX). Second, we have the watch list section which contains the stocks, equities, or the securities that the user wants to keep track. The indicators, as discussed in the technology section, uses a two-week time interval to predict how the stocks in the watch list are going to perform, based on our models for the future 7 days. The user after seeing our prediction can then go on to do more research using other indicators such as the stochastic oscillator, MACD, or the momentum for the stock, to determine a more technical analysis. The user using our software can judge the future performance of their portfolio using strictly the **Volatility** Index and moving average flags as indicated by the **volatility** index or another indicator added. Figures 11 and 12 below show the regression tool component of our web application and a bit more detailed page about the stock in the watch list, respectively.

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