High frequency vs. daily resolution: The economic value of forecasting volatility models 2nd ed

Lilla, Francesca Working Paper High frequency vs. daily resolution: The economic value of forecasting volatility models 2nd ed Quaderni - Working Paper DSE, No. 1099 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lilla, Francesca (2017) : High frequency vs. daily resolution: The economic value of forecasting volatility models 2nd ed, Quaderni - Working Paper DSE, No. 1099, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/5541 This Version is available at: https://hdl.handle.net/10419/177599 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ ISSN 2282-6483 High Frequency vs. Daily Resolution: the Economic Value of Forecasting Volatility Models 2nd ed Francesca Lilla Quaderni - Working Paper DSE N°1099 High Frequency vs. Daily Resolution: the Economic Value of Forecasting Volatility Models* Francesca Lilla † April 9, 2017 Abstract Forecasting volatility models typically rely on either daily or high frequency (HF) data and the choice between these two categories is not obvious. In particular, the latter allows to treat volatility as observable but they suffer from many limitations. HF data feature microstructure problem, such as the discreteness of the data, the properties of the trading mechanism and the existence of bid-ask spread. Moreover, these data are not always available and, even if they are, the asset’s liquidity may be not sufficient to allow for frequent transactions. This paper considers different variants of these two family forecasting-volatility models, comparing their performance (in terms of Value at Risk, VaR) under the assumptions of jumps in prices and leverage effects for volatility. Findings suggest that daily-data models are preferred to HF-data models at 5% and 1% VaR level. Specifically, independently from the data frequency, allowing for jumps in price (or providing fat-tails) and leverage effects translates in more accurate VaR measure. JEL-Classification: C58 C53 C22 C01 C13 Keywords: GARCH, DCS, jumps, leverage effect, high frequency data, realized variation, range estimator, VaR *A previous version, with different results but with the same title, circulated in 2016, as Quaderni - Working Paper DSE N. 1084 of the Department of Economics. University of Bologna †University of Bologna, Department of Economics; Piazza Scaravilli 2, 40126 Bologna, Italy. E-mail: [email protected] 1 1 Introduction Modeling and forecasting volatility of asset returns are crucial for many applications, such as asset pricing model, risk management theory and portfolio allocation decisions. An earlier literature, including Engle (1982) and Bollerslev (1986) among others, has developed models of asset volatility dynamics in discrete time, known as heteroscedastic volatility models, i.e. ARCH-GARCH. Thanks to the availability of high frequency (HF) data, a new strand of literature has originated a new class of models based on the Realized Volatility (RV) estimator, therefore introducing a non-parametric measure of return volatility (see Andersen et al.,001a,Barndorff-Nielsen,2002 and Andersen et al.,2012). As thw main innovation, RV models provides an ex-post observation of volatility, at odds with the standard ARCHGARCH approach, that treats volatility as a latent variable. Although forecasting-volatility models based on HF data are getting more and more popular in the literature, the choice between HF-data and daily-data models is yet not obvious, in particular from an applied standpoint. In particular, the former still suffer from various limitations, that can be addressed only at the cost of a heavy manipulation of the original data. One of the main issues is the presence of the market microstructure noise, which prevents from getting a perfect estimate (at the limit) of the returns’ variance (see Hansen and Lunde, 2006 and A¨ ıt-Sahalia et al.,2005,2011). The market microstructure noise may originate from different sources, including the discreteness of the data, the properties of the trading mechanisms and the existence of a bid-ask spread. Regardless of the source, when return from assets are measured based on their transaction prices over very tiny time intervals, these measures are likely to be heavily affected by the noise and therefore brings little information on the volatility of the price process. Since the level of volatility is proportional to the time interval between two successive observations, as the time interval increases, the incidence of the noise remains constant, whereas the information about the ”true” value of the volatility increases. Therefore, there is a trade-off between high frequency and accuracy, which has led authors to identify an optimal sampling frequency of 5 minutes1. 1Since the best remedy for market microstructure noise depends on the properties of the noise, if data sampled at higher frequency, e.g. tick-by-tick, are used the noise term needs to be modeled and, as far as I know, there is no unified framework about how to deal with it. A¨ ıt-Sahalia et al. (2005) define a new estimator, Two Scales Realized Volatility (TSRV), which takes advantages of the rich information of tick-by-tick data and corrects the 2 HF data also features another inconvenient: they are not always available and, even if they are, the asset may be not liquid enough to be frequently traded. On the contrary, daily data are relatively simple to record and collect and are commonly easy-to-get. This paper sheds light on the choice between HF-data and daily data models, by assessing the economic value of the two family models, based on a comparison of their performance in forecasting asset volatility. Following the risk management perspective, I use value at risk (VaR) as the econometric metric of volatility forecastability, as suggested by Christoffersen and Diebold (2000). VaR is defined as the quantile of the conditional portfolio distribution, and is therefore quite intuitive as a measure: indeed, it is the most popular quantitative measure of the market risk associated with a portfolio of assets, and is generally adopted by banks and required by regulators all over the world2. In running the comparison between HF-data and daily data models, this paper introduces two key assumptions. Firstly, the data generating process for asset prices features discontinuities in its trajectories, jumps3. Secondly, volatility (i.e. the standard deviation of asset return) reacts differently to changes in asset return which have the same magnitude, but different sign, leverage effect. These two assumptions represent the main novelty of this paper since none of the previous studies on the economic value of different forecasting-volatility models has investigated the matter under both jumps in price and leverage effect combined effects of microstructure noise on volatility estimation. The authors, instead of sampling over a longer time horizon and discarding observations, make use of all data and model the noise as an ”observation error”. But the microstructure noise modeling goes beyond the scope of this work. 2Banks often construct VaR from historical simulation (HS-VaR): VaR is the percentile of the portfolio distribution obtained using historical asset prices and today weights. This procedure is characterized by a slow reaction to market conditions and for the inability to derive the term structure of VaR. The VaR term structure explains how risk measures vary across different investment horizons. In HS-VaR, for example, if T-day 1% VaR is calculated, the 1-day 1% VaR is simply scaled by √T. This relation is valid only if daily returns are i.i.d. realizations of a Normal distribution. We know that is not the case since returns present leptokurtosis and asymmetry. The main limit of HS-VaR is the substitution of the conditional return distribution with the unconditional counterpart. Risk Metrics and GARCH models represent improvements over HS-VaR measure. Both of them provide an explicit assumption about the DGP and the conditional variance but they have also important differences. In addition to the estimation method: GARCH conditional volatility is estimated by maximizing the log-likelihood function while the parameters used in Risk Metrics are chosen in an ad hoc fashion, they differ for the possibility to account for the term structure of VaR. This is because GARCH process allows for mean reversion in volatility while Risk Metrics does not, reproducing a flat term structure for VaR. 3A continuous price process is a restrictive assumption since it is not possible to distinguish between the dynamic originated from the two sources of variability, i.e. continuous and discontinuous movements with consequences on the return generating process 3 together. Giot and Laurent (2004) compare the performance of a daily ARCH-type model with the performance of a model based on the daily RV in a VaR framework. The authors find that VaR specification based on RV does not really improve the performance of a VaR model estimated using daily returns. This paper underlines an important issue: in economics applications, it is important to recognize and take into account the key features of the empirical data in order to choose a valid data generating process. Clements et al. (2008) evaluate quantile forecasts focusing exclusively on models based on RV in order to understand if the results presented for stock returns can be carried over exchange rates. According to the results in Clements et al. (2008) the distributional assumption for expected future returns is needed for computing quantile, irrespective of the frequency of data used. Brownlees and Gallo (2010) forecast VaR using different volatility measures based on ultra-high-frequency data using a two-step VaR prediction procedure. They find that using ultra-high-frequency observations, VaR predictive ability is considerably improved upon relative to a baseline GARCH but not so relative to the range. The reason is related to the microstructure noise issue which arises when ultra high-frequency data are used. Indeed I want to contribute to the existing literature focusing on the measurement and the efficient use of the information embedded in HF data with respect to the information content of daily observations. Assuming both jumps and leverage effects in the returns dynamics for both data categories, I provide a more balanced comparison than in the previous work. In the choice of the model to use for the comparison, I consider the GARJI model of Maheu and McCurdy (2004), as the baseline for the daily data models. The latter is a mixed-GARCH jump model which allows for asymmetric responses to past innovations in asset returns: the news impact (resulting in jump innovations) may have a feedback effect on the expected volatility, in addition to the feedback effect associated with the normal error term. For the case of HF data, I consider models in which Realized Volatility (RV) is decomposed into continuous and discontinuous volatility components. The continuous component is captured by means of the bi-power variation (BV), introduced by Barndorff-Nielsen and Shephard (2004), whereas the discontinuous component (JV) is obtained as the difference between RV and BV 4 at given point in time4. In Andersen et al. (2007), JV is obtained considering only jumps that are found to be significative, and neglecting the others5.Corsi et al. (2010) consider instead all jumps, stressing the importance to correct the positive bias in BV due to jumps classified as consecutive. In this paper, I consider both these approaches and make a comparison among them, finding evidence in favor of jump identification strategy of Corsi et al. (2010) when the leverage effect is introduced. To account for the leverage effect, I introduce in this class of models the heterogeneous structure proposed by Corsi and Ren´ o(2009). Throughout this paper, the GARJI-VaR measures are obtained by following Chiu et al. (2005), that is, by adjusting for skewness and fat tails in the specification of the conditional distribution of returns6. The HF-VaR measures, instead, are computed by assuming a conditional Gaussian distribution for asset returns: as shown in Andersen et al. (2010), returns standardized for the square root of RV are indeed approximatively Normal7. In order to assess the model’s capability to forecast future volatility, I implement a backtesting procedure based on both the Christoffersen (1998) test and the Kupiec (1995) test. In addition to comparing the economic value of daily data and HF-data models, the analysis performed in this paper sheds light on three other issues. The first is represented by the economic value per se, i.e. out of the comparison, of the class of forecasting volatility models adopting HF-data. This is done by considering different specifications of this family models. I first run a comparison among them (based on their forecasting performances); then, I compare some of them with their variant, obtained by using the Range estimator (RA) of Parkinson (1980). The choice of this particular benchmark is motivated by the fact that the RA estimator is likely to deliver a measure of volatility which lies in the middle of the mea- 4As shown in Andersen et al. (2002), Andersen et al. (2007), RV is a consistent estimator for the quadratic variation, whereas BV represents a consistent estimator of the continuous volatility component, i.e. the so-called integrated volatility, in the presence of jumping prices. 5The authors with significant jumps refer to large value of RVt−BVtwhile small positive values are treated both as part of continuous sample path variation or as measurement errors. 6The computation of VaR measure requires, in addition to the conditional volatility dynamics, the specification of the conditional distribution of returns.VaR is a conditional risk measure so an assumption on the conditional distribution of returns is needed. Conditional normality is an acceptable assumption (returns standardized by their conditional volatility could be approximately Gaussian even if the unconditional returns are not Gaussian) only if the volatility model is able to fatten conditionally Gaussian tails enough to match the unconditional distribution. If this is not the case another conditional distributional assumption is necessary. 7This result is confirmed by the standardized returns of the sample used in this paper. See Section 2. 5 sure obtained from HF estimators and that obtained from daily data models8. My findings suggest that HF-data models which explicitly provide both jumps and leverage factors stand out from the others in term of forecasting capability. The second by-product of my analysis is a quantitative assessment of the importance of the explicit jump component in the conditional distribution of asset returns 9. This point is addressed in both the family models considered in this paper. Hence, I first compare the forecasting volatility performances of each HF-data model with and without a decomposition of the RV into the continuous and the discontinuous component. Then, I run a similar analysis for the case of the daily data models, considering the GARCH-t model, as well as the Beta-t model10 proposed by Harvey and Luati (2014). According to my analysis, introducing an explicit, persistent jump component in the conditional return dynamics (together with an asymmetric response to bad and good news into conditional volatility dynamics) may help to forecast the ex-post volatility dynamics and obtain more accurate VaR measures, at least at the VaR level required by Basel accords (1%). For HF-data models, accounting for jumping prices does not seem to improve significantly the accuracy of the estimates. The last issue of my analysis is related to the importance of leverage effect in forecasting volatility. The findings in this paper recommend the explicit introduction of a factor that generates the asymmetric volatility response to price movements in the forecasting model. The rest of the paper is organized as follows. Section 2summarizes the volatility measures and the forecasting models based on both HF and daily data. Section 3and Section 4show, respectively, the backtesting methods used to evaluate forecasting models accuracy and the empirical results. Section 5concludes. 8The RA estimator exploits information on the highest and the lowest price recorded in a given day for a particular asset. In this respect, it requires information on the intra-day activity (going beyond the simple closing price of the asset), but without relying on further information, that might be not readily available). 9The presence of a jump component is justified both at theoretical and empirical level. From a theoretical perspective, an explicit discontinuous volatility-component allows to have information on the market response to outside news, which is key for many applications. From an empirical standpoint, instead, it is very difficult to distinguish power-type tails from exponential-type tails, given that is not clear to what extent the return distribution is heavily tailed. In this regard, the jump component of a jump-diffusion model may be interpreted as the market response to outside news: when good or bad news arrive at a given point in time, the asset price changes according to the jump size (and the jump sign) and an extreme sources of variation is added to the idiosyncratic component. 10Beta-tmodel, belongs to the general class of Dynamic Conditional Score (DCS) model. They are also known as Generalized Autoregressive Score (GAS) model proposed by Creal et al. (2013). 6 2 Volatility Measures and Forecasts 2.1 Estimates of volatility with High Frequency Data The RV measure is an estimator for the total quadratic variation, namely, it converges in probability, as the sampling frequency increases, to the continuous volatility component if there are no jumps. Instead, it converges to the sum of continuous and discontinuous volatility components if at least one jump occurs. As explained in Andersen et al. (2012), it is possible to use the daily RV measures, the ex-post volatility observations, to construct the ex-ante volatility forecasts. This is possible simply by using standard ARMA time series tools but it is important to take into account the difference with GARCH-type forecasting. The fundamental difference is that in the former case the risk manager treats volatility as observed while in the latter framework volatility is inferred from past returns conditional on a specific model. The idea behind the RV is the following: even if prices are not available on continuous basis, prices are recorded at higher frequency than daily. Using these squared returns a daily RV could easily be computed. In this way the ex-post volatility is considered as observable at each point in time. More precisely, the RV on day tbased on returns at the ∆intraday frequency is RVt(∆)≡ N(∆) ∑ j=1 r2 t,j where rt,j=pt−1+j∆−pt−1+(j−1)∆and pt−1+j∆is the log-price at the end of the jth interval on day tand N(∆)is the number of the observations available at day trecorded at ∆frequency. In the absence of microstructure noise, as ∆→0 the RV estimator approaches the integrated variance of the underlying continuous-time stochastic volatility process on day t: RVt−→pIVtwhere IVt=Zt t−1σ2(τ)dτ Furthermore, in this paper I assume that the the underlying price process is characterized by discontinuities. Indeed, the previous convergence is not valid but the RV estimators approaches in probability to the sum of the integrated volatility and the variation due to jumps 7 novations and then obtain a richer characterization of volatility dynamics, especially with respect to events in the tail of the distribution (jumps). In particular E[Nt−1|Ft−1)is the ex-post assessment of the expected number of jumps that occurred from t−2 to t−1 and it is equal to ∑∞ j=0jP(Nt−1=j|Ft−1). Therefore ξt−1is the change in the econometrician’s conditional forecast on Nt−1as the information set is updated, it is the difference between the expected value and the actual one. As shown by Maheu and McCurdy (2004) this expression may be inferred using Bayes’ formula: P(Nt=j|Ft−1) = f(Rt|Nt=j,Ft−1)P(Nt=j|Ft−1) f(Rt|Ft−1)for j=0, 1, 2, . . . (20) Indeed, conditional on knowing λt,σt, and the number of jumps that took place over a time interval, Nt=j, the density of Rtin terms of observable is Normal: f(Rt|Ft−1) = ∞ ∑ j=0 f(Rt|Nt=j,Ft−1)×P(Nt=j|Ft−1)(21) where f(Rt|Nt=j,Ft−1) = 1 q2π(σ2 t+jδ2) exp −(Rt−µ+θλt−θj)2 2(σ2 t+jδ2)(22) Naturally the likelihood function is defined starting from (22), where ˜ θis the vector of the parameters of interest, i.e. ˜ θ= (γ,ρ,θ,δ2,α,αj,αa,αaj,ω,β,λ0,µ): L(Rt|Nt=j,Ft−1;˜ θ) = T ∏ t=1 f(Rt|Nt=j,Ft−1)(23) and the log-likelihood is: l(Rt|Nt=j,Ft−1;˜ θ) = T ∑ t=1 log f(Rt|Nt=j,Ft−1)(24) The maximum number of jumps in each day in the filter (20) is set equal to 10. This is becasue, as suggested in Maheu and McCurdy (2004), the conditional Poisson distribution has almost zero probability in the tails for values of Nt≥10. In order to isolate the role of jumps, I estimate a nested version of the GARJI model, i.e. ARJI, which is obtained by imposing αj=αa=αa,j=0. 14 In addition, I consider the GARCH-tmodel and Beta-t-GARCH model for conditional volatility. The aim is to understand if the ARJI model can provide a better fit to the empirical distribution of the data and a better quantile forecast with respect to volatility specifications based on fat tails, such as t-Student. In particular, Beta-t-GARCH presents a more sophisticated volatility specification with respect to GARCH-tmodel. The former consists of an observation driven model based on the idea that the specification of the conditional volatility as a linear combination of squared observations is taken for granted but, as a consequence, it responds too much to extreme observations and the effect is slow to dissipate. Harvey and Luati (2014) define a class of models (DCS) in which the observations are generated by a conditional heavy-tailed distribution with time-varying scale parameters and where the dynamics are driven by the score of the conditional distribution. In this way, Beta-t-GARCH counts the innovation outliers but also the additive outliers. 3 Computing and comparing VaR forecasts The VaR is defined as the 100α% quantile of the distribution of returns. The probability that the return of a portfolio over a tholding period will fall below the VaR is equal to 100α%. The predicted VaRs are based on the predicted volatility and they depend on the assumption on the conditional density of daily returns. The one day-ahead VaR prediction at time t+1 conditional on the information set at time tis: d VaRt+1|t=qb σ2 t+1|tF−1 t(α)(25) In (25)b σ2 t+1|tis the returns variance, estimated in both parametric and non-parametric models, F−1 t(α)is the inverse of the cumulative distribution of daily returns while αindicates the degree of significance level. In the case of HF data b σ2 t+1|tis equal to c RVtor d RAtestimated as explained in the section 2.2 while for GARJI model the returns variance is not simply the modified GARCH dynamic but it also consist of the variance due to jumps (Hung et al., 2008): d VaRt+1|t=qb σ2 t+1|t+ (b θ2 t+b δ2 t)b λte F−1 t(α)(26) 15 where e F−1 t(α) = F−1 t(α) + 1 6((F−1 t(α))2−1)Sk(Rt|tFt−1)and Sk(Rt|tFt−1)is the conditional return skewness computed after estimating the model. Once obtained VaR forecasts, I assess the relative performance of the models through the violation16 rate and the quality of the estimates by applying backtesting methods17. A violation occurs when a realized return is greater than the estimated ones (VaR). The violation rate is defined as the total number of violations divided by the total number of one period-forecasts18The tests used in this paper are the Unconditional Coverage (LUC) and Conditional Coverage (LCC) tests suggested respectively by Kupiec (1995) and Christoffersen (1998). The LUCand LCC are the most popular tests among practitioners and academics. This is because they are very simple to implement and because they are incorporated in the Basel accords requirements 19. These two motivations represent also the reason why both tests are used also in the academic literature. The LUC and the LCCtests assess the adequacy of the model by considering the number of VaR exceptions, i.e. days when returns exceed VaR estimates. If the number of exceptions is less than the selected significance level would indicate, the system overestimates risk; on the contrary too many exceptions signal underestimation of risk. In particular, the first test examines whether the frequency of exceptions over some specified time interval is in line with the selected significance level. A good VaR model produces not only the “correct” amount of exceptions but also exceptions that are independent each other and, in turn, not clustered over time. The test of conditional coverage takes into account for the number of exceptions and when the exceptions occur. The tick loss function considered is defined as Binary loss function (BLF) which counts the number of exceptions, that are verified when the loss is larger than the forecasted VaR: 16In the testing literature exception is used instead of violation because the former is referred, as I explain later, to a loss function. The loss function changes according to the test applied and the motivation behind the testing strategies. 17The backtesting tests give the possibility to interpret the results and then the quality of the forecasting model choose in inferential terms. 18As well explained in Genc¸ay et al. (2003) atqth quantile, the model predictions are expected to underpredict the realized return α= (1−q)percent of the time. A high number of exceptions implies that the model excessively underestimates the realized return. If the exception ratio at the q th quantile is greater than αpercent, this implies excessive underprediction of the realized return. If the number of exceptions is less than αpercent at the q th quantile, there is excessive overprediction of the realized return by the underlying model. 19See Nieto and Ruiz,2016 for a review on VaR forecasting and evaluation through backtesting. 16 BLFt+1=     1 if Rt+1<d VaRt+1|t 0 if Rt+1≥d VaRt+1|t (27) where d VaRt+1|tis the estimated VaR at time tthat refers to the period t+1. The Likelihood Ratio test of unconditional coverage tests the null hypothesis that the true probability of occurrence of an exception over a given period is equal to α: H0:p=α H1:p6=α where b p=n0 n1+n0is the unconditional coverage (the empirical coverage rate) or the failure rate and n0and n1denote, respectively, the number of exceptions observed in the sample size and the number of non-exceptions. The unconditional test statistic is given by: LRUC =−2 log (1−α)n1αn0 (1−b p)n1b pn0∼χ2(1)(28) So, under the null hypothesis the significance level used to forecast VaRs and the empirical coverage rate are equal. The test of conditional coverage proposed by Christoffersen (1998) is an extended version of the previous one taking into consideration whether the probability of an exception on any day depends on the exception occurrence in the previous day. The loss function in constructed as in (27) and the log-likelihood testing framework is as in (28) including a separate statistic for the independence of exceptions. Define the number of days when outcome joccurs given that outcome ioccurred on the previous day as nij and the probability of observing an exception conditional on outcome iof the previous day as πi. Summarizing: π0=n01 n00 +n01 π1=n11 n10 +n11 π=n01 +n11 n00 +n01 +n10 +n11 (29) 17 The independence test statistic is given by: LRIND =−2 log (1−π)n00+n10 πn01+n11 (1−π0)n00 πn01 0(1−π1)n10 πn11 1(30) Under the null hypothesis the first two probabilities in (29) are equal, i.e. the exceptions do not occur in cluster. Summing the statistics (28) and (30) the conditional coverage statistic is obtained, i.e. LRCC =LRUC +LRIND and it is distributed as a χ2with two degrees of freedom since two is the number of possible outcomes in the sequence in (27). In order to avoid the possibility that the models considered passing the joint test but fail either the coverage or the independence test I choose to run LRCC and also its decomposition in LRUC and LRIND. 4 Data and Empirical results 4.1 Data In order to assess the informational content of HF and daily data, I use S&P 500 index from 5 Jan.1996 to 30 Dec.2005 for both samples. The total number of trading days is equal to 2516 which coincides with the number of daily returns. In the top panel of Figure 1the level of the S&P 500 index is presented. The corresponding daily returns are displayed in the bottom panel of Figure 1. Given the literature on the effects of microstructure noise of estimates of RV and the forecast performance of RV models based on different sampling frequency, I use 5-minutes data for a total of 197, 689 observations. I compute 5-minutes intraday returns as the log-difference of the closing prices in two subsequent periods of time. The daily returns are computed taking the last closing prices in each trading day. The range volatility at each date is calculated as scaled log difference between the highest and the lowest price in a trading day. Table 1reports the descriptive statistics of S&P 500 index for RAt,RVtand its decomposition in BVtand JVt. In particular JVtis computed as max{RVt−BVt, 0}20. A number of interesting features are founded. Firstly, returns exhibit negative asymmetry and leptokurtosis. As shown in Ander- 20The summary statistics of the continuous and disontinuous components computed according to Andersen et al. (2007) and Corsi et al. (2010) are not reported because are very similar to those presented in Table 1. 18 Figure 1: Top: daily S&P 500 index from 5 Jan.1996 to 30 Dec.2005. The horizontal axis corresponds to time while the vertical axis displays the value of the index. Bottom: daily S&P 500 percentage returns calculated by rt=log(pt/pt−1), where ptis the value of the index at time t. Table 1: Summary Statistics. The rows report the sample mean, standard deviation, skewnwss, kurtosis, sample minimum and maximum for the daily returns (Rt), the standardized daily returns (Rt/√RVt) the daily realized volatility (RVt), the daily bipower variation (BVt), the daily jump component (JVt) and the daily range estimator (RAt). Returns are expressed in percentage. RtRt/√RVtRVtBVtJVtRAt Mean 0.0279 0.1378 0.8250 7.93E-05 3.15E-06 9.70E-05 St. Dev. 1.1520 1.3138 1.0097 9.85E-05 1.00E-05 1.52E-04 Skewness -0.0951 0.253 4.8721 4.8786 19.9283 7.1671 Kurtosis 5.9165 2.8505 39.1013 39.3401 659.3967 84.1687 Min -7.1127 -3.6092 0.0281 0.0281 0 0.0206 Max 5.3080 4.7161 11.890 11.890 3.6200 25.931 sen et al. (2007) the daily returns standardized with respect to the square root of the ex-post realized volatility are closed to Gaussian. In fact its mean and asymmetry are close to zero, 19 its variance is close to one while its kurtosis is near to 3. This result is clear from Figure 2 in which the empirical density distribution is plotted with the normal density distribution for Rt/√RVt. Moreover if I compare RVtand BVtthe latter is less noisy than the former, considering the role of jumps. Finally, jump process does not show any Gaussian feature 21. Figure 2: Standardized log-returns distribution of the S&P 500 index.The standard normal distribution (solid line) is compared with the standardized log-returns distribution (dashed line). Figure 3shows the plot of RVt,BVt,JVtand RAtestimators. It is evident that RVt,BVt and JVtfollow a similar pattern and the latter tends to be higher when RVtis higher. JVt exhibits a relatively small degree of persistence as consequence of the clustering effect. Not surprisingly, RAtfollows the same pattern of RVtsince both of them are ex-post volatility measures. 4.1.1 Estimation results based on daily data Table 2, provides parameter estimates for both the GARJI and ARJI model applied to the S&P500. The parameter estimates are presented separating the diffusion component from 21In particular, jumps computed according to (6) exhibit a higher mean with respect to those computed according to (4), given that the former exploits the possibility of consecutive jumps. 20 Figure 3: Top: RVtcomputed using 5- minutes data from 5 Jan.1996 to 30 Dec.2005. Second: BVtcomputed using 5- minutes data from 5 Jan.1996 to 30 Dec.2005. Third: JVt= max{RVt−BVt, 0}is computed using 5- minutes data from 5 Jan.1996 to 30 Dec.2005. Bottom: Range estimator computed using daily data from 5 Jan.1996 to 30 Dec.2005. Time is on the horizontal axis. 21 the jump component. First, both parameters ρand γare significantly different from zero. The former represents the persistence of the arrival process of jumps that is quite high for both models implying the presence of jump clustering. The latter, γ, measures the change in the conditional forecast of the number of jumps due to the last day information. The significance of these two parameters suggests that the arrival process of jumps can deviate from its unconditional mean. The implied unconditional jump intensity is 0.8727 while the average variance due to jumps is equal to 0.5516: the index is volatile. This result is confirmed by the average proportion of conditional variance explained by jumps which is equal to 0.3068, jumps explained almost the 23% of the total returns variance. Moreover the jump size mean θ is negative for both model and the most interesting feature is that it affects conditional skewness and conditional kurtosis. The sign of θindicates that large negative return realizations due to jumps are associated with an immediate increase in the variance explaining the contemporaneous leverage effect: when jumps are realized they tend to have a negative effect on returns. In particular the average conditional skewness is equal to −0.2766 while the average conditional kurtosis is equal to 3.2814. Furthermore the feedback coefficient g(Λ,Ft−1)tends to be smaller when at least one jump occurs because the total innovation is larger after jumps . Considering the first column of Table 2, the feedback coefficient associated with good news and no jump is equal to 0.0005 and it increases if one jump occurs, i.e. 0.0010. If no jumps occur and if news are bad the coefficient is equal to 0.0411; it is equal to 0.0348 in case of bad news if one jump occurs. These results provide evidence for the asymmetric effect of good and bad news and they show that the asymmetry associated to bad news is more important in the absence of jumps, namely for normal innovations. In fact the difference between the coefficient estimates for both good and bad news in the case of no jumps and one jump are quite similar. This means that news associated with jump innovations is incorporated more quickly into current prices. The second column of Table 2presents the estimated parameters for the model with αj=αa=αa,j=0. With this specification and through the LR test it is possible to understand if the asymmetric effect of good versus bad news is statistically significant: the asymmetric news effect is statistically significant. 22 Table 2: GARJI and ARJI models estimates. ARJI model is obtained assuming αj=αa= αa,j=0. Standard errors are in parenthesis. Process Parameters S&P 500 GARJI ARJI Diffusion µ0.0106 (1.9839) 0.0153 (2.1142) ω0.0036 (0.0005) 0.0034 (0.0005) α-7.7048 (0.4332) -4.7623 (0.3063) αj0.8096 (0.7538) - αa4.5131 (0.4213) - αa,j-0.9776 (0.7204) - β0.9696 (0.0002) 0.9787 (0.0000) Jump λ00.0211 (0.0039) 0.0229 (0.0052) ρ0.9758 (0.0025) 0.9757 (0.0030) γ0.5262 (0.0501) 0.4792 (0.0641) θ-0.9895 (0.3985) -0.9793 (0.4501) δ20.0005 (0.0000) 0.0000 (0.0000) Log-likelihood -3570.8 -3574.4 4.1.2 Estimation results based on high frequency data All the estimates presented in Table 3, Table 4and Table 5are computed employing the OLS method over the entire sample period, i.e. from 5 Jan. 1996 to 30 Dec. 2005, for the S&P500 index. Table 3and Table 4show the results for the models presented in Section 2.2 for models based on RV, its decomposition in BV and JV and the cascade structure for the leverage effect. The coefficients of the continuous component expressed as daily, weekly and monthly measures, respectively β1,β2and β3are significants in all models. Moreover, jump components appear to be fundamental to forecast one step ahead volatility; the predictive power is larger for those specifications that allow for RV decomposed in its continuous and discontinuous components, regardless the identified method used for jump magnitude. Furthermore, the estimates for the aggregate leverage variables are negatives (as expected) and significant. Moreover, the predictive power increases adding the cascade structure for the leverage 23 Appendix A VaR accurancy at 10% level Table 8: VaR accurancy at 10% level. The first column shows the model chosen in order to compute the VaR forecasts. H is the average number of violations computed for each model. VaR is the average VaR forecasts. LRUC, LRCC and LRIND represent the pvalue associated to the Kupiec (1995) and Christoffersen (1998) tests. All tests are evaluated at 1% significance level. Model H VaR LRUC LRIND LRCC AR(8) 0.132 -1.276 0.003 0.670 0.010 HAR 0.139 -1.267 0.000 0.817 0.001 L-HAR 0.136 -1.268 0.001 0.881 0.004 HARC-Jumps 0.142 -1.261 0.000 0.945 0.001 LHARC-Jumps 0.138 -1.266 0.000 0.987 0.002 HAR-CV-JV 0.142 -1.260 0.000 0.726 0.001 LHAR-CV-JV 0.137 -1.265 0.001 0.949 0.003 HAR-C-J 0.147 -1.257 0.000 0.979 0.000 LHAR-C-J 0.141 -1.261 0.000 0.913 0.001 GARJI 0.070 -1.646 0.003 0.165 0.004 ARGJI 0.068 -1.673 0.001 0.279 0.003 GARCH-t 0.052 -2.221 0.000 0.029 0.000 Beta-t-GARCH 0.052 -2.195 0.000 0.029 0.000 Range AR(8) 0.143 -1.255 0.000 0.559 0.000 Range HAR 0.145 -1.233 0.000 0.621 0.000 Range L-HAR 0.166 -1.135 0.000 0.439 0.000 30 References A¨ ıt-Sahalia, Y. and J. Jacod (2010). Analyzing the spectrum of asset returns: Jump and volatility components in high frequency data. Technical report, National Bureau of Economic Research. A¨ ıt-Sahalia, Y. and L. Mancini (2008). Out of sample forecasts of quadratic variation. Journal of Econometrics 147(1), 17–33. A¨ ıt-Sahalia, Y., P. A. Mykland, and L. Zhang (2005). How often to sample a continuous-time process in the presence of market microstructure noise. Review of Financial studies 18(2), 351–416. A¨ ıt-Sahalia, Y., P. A. Mykland, and L. Zhang (2011). Ultra high frequency volatility estimation with dependent microstructure noise. Journal of Econometrics 160(1), 160–175. Alizadeh, S., M. W. Brandt, and F. X. Diebold (2002). Range-based estimation of stochastic volatility models. The Journal of Finance 57(3), 1047–1091. Andersen, T. G. and T. Bollerslev (1998). Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International economic review, 885–905. Andersen, T. G., T. Bollerslev, P. F. Christoffersen, and F. X. Diebold (2012). Financial risk measurement for financial risk management. Technical report, National Bureau of Economic Research. Andersen, T. G., T. Bollerslev, and F. X. Diebold (2007). Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility. The Review of Economics and Statistics 89(4), 701–720. Andersen, T. G., T. Bollerslev, F. X. Diebold, and H. Ebens (2001a). The distribution of realized stock return volatility. Journal of financial economics 61(1), 43–76. Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys (2003). Modeling and forecasting realized volatility. Econometrica 71(2), 579–625. 31 Andersen, T. G., T. Bollerslev, F. X. Diebold, and C. Vega (2002). Micro effects of macro announcements: Real-time price discovery in foreign exchange. Technical report, National bureau of economic research. Andersen, T. G., T. Bollerslev, P. Frederiksen, and M. Ørregaard Nielsen (2010). Continuoustime models, realized volatilities, and testable distributional implications for daily stock returns. Journal of Applied Econometrics 25(2), 233–261. Barndorff-Nielsen, O. E. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 64(2), 253–280. Barndorff-Nielsen, O. E. and N. Shephard (2004). Power and bipower variation with stochastic volatility and jumps. Journal of financial econometrics 2(1), 1–37. Bates, D. S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of financial studies 9(1), 69–107. Bates, D. S. (2000). Post-’87 crash fears in the s&p 500 futures option market. Journal of Econometrics 94(1), 181–238. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of econometrics 31(3), 307–327. Bollerslev, T. and V. Todorov (2011). Tails, fears, and risk premia. The Journal of Finance 66(6), 2165–2211. Brownlees, C. T. and G. M. Gallo (2010). Comparison of volatility measures: a risk management perspective. Journal of Financial Econometrics 8(1), 29–56. Chan, W. H. and J. M. Maheu (2002). Conditional jump dynamics in stock market returns. Journal of Business & Economic Statistics 20(3), 377–389. Chiu, C.-L., M.-C. Lee, and J.-C. Hung (2005). Estimation of value-at-risk under jump dynamics and asymmetric information. Applied Financial Economics 15(15), 1095–1106. Christensen, K., R. C. Oomen, and M. Podolskij (2014). Fact or friction: Jumps at ultra high frequency. Journal of Financial Economics 114(3), 576–599. 32 Christoffersen, P., R. Elkamhi, B. Feunou, and K. Jacobs (2009). Option valuation with conditional heteroskedasticity and nonnormality. Review of Financial studies, hhp078. Christoffersen, P. F. (1998). Evaluating interval forecasts. International economic review, 841– 862. Christoffersen, P. F. and F. X. Diebold (2000). How relevant is volatility forecasting for financial risk management? Review of Economics and Statistics 82(1), 12–22. Clements, M. P., A. B. Galv˜ ao, and J. H. Kim (2008). Quantile forecasts of daily exchange rate returns from forecasts of realized volatility. Journal of Empirical Finance 15(4), 729–750. Corsi, F. (2009). A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics, nbp001. Corsi, F., D. Pirino, and R. Reno (2010). Threshold bipower variation and the impact of jumps on volatility forecasting. Journal of Econometrics 159(2), 276–288. Corsi, F. and R. Ren´ o (2009). Har volatility modelling with heterogeneous leverage and jumps. Available at SSRN 1316953. Creal, D., S. J. Koopman, and A. Lucas (2013). Generalized autoregressive score models with applications. Journal of Applied Econometrics 28(5), 777–795. Diebold, F. X. and R. S. Mariano (2012). Comparing predictive accuracy. Journal of Business & economic statistics. Duan, J.-C. et al. (1995). The garch option pricing model. Mathematical finance 5(1), 13–32. Duan, J.-C., P. Ritchken, and Z. Sun (2006). Approximating garch-jump models, jumpdiffusion processes, and option pricing. Mathematical Finance 16(1), 21–52. Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica: Journal of the Econometric Society, 987–1007. Engle, R. F. and S. Manganelli (2004). Caviar: Conditional autoregressive value at risk by regression quantiles. Journal of Business & Economic Statistics 22(4), 367–381. 33 Genc¸ay, R., F. Selc¸uk, and A. Ulug¨ ulyaˇ gci (2003). High volatility, thick tails and extreme value theory in value-at-risk estimation. Insurance: Mathematics and Economics 33(2), 337–356. Ghysels, E., P. Santa-Clara, and R. Valkanov (2004). The midas touch: Mixed data sampling regression models. Finance. Giot, P. and S. Laurent (2004). Modelling daily value-at-risk using realized volatility and arch type models. journal of Empirical Finance 11(3), 379–398. Hansen, P. R. and A. Lunde (2006). Realized variance and market microstructure noise. Journal of Business & Economic Statistics 24(2), 127–161. Harvey, A. and A. Luati (2014). Filtering with heavy tails. Journal of the American Statistical Association 109(507), 1112–1122. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of financial studies 6(2), 327–343. Heyde, C., S. Kou, and X. Peng (2006). What is a good risk measure: Bridging the gaps between data, coherent risk measures and insurance risk measures. Preprint, Columbia University. Huang, X. and G. Tauchen (2005). The relative contribution of jumps to total price variance. Journal of financial econometrics 3(4), 456–499. Hung, J.-C., M.-C. Lee, and H.-C. Liu (2008). Estimation of value-at-risk for energy commodities via fat-tailed garch models. Energy Economics 30(3), 1173–1191. Kupiec, P. H. (1995). Techniques for verifying the accuracy of risk measurement models. THE J. OF DERIVATIVES 3(2). Maheu, J. M. and T. H. McCurdy (2004). News arrival, jump dynamics, and volatility components for individual stock returns. The Journal of Finance 59(2), 755–793. Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of financial economics 3(1), 125–144. 34 M¨ uller, U. A., M. M. Dacorogna, R. D. Dav´ e, R. B. Olsen, O. V. Pictet, and J. E. von Weizs¨ acker (1997). Volatilities of different time resolutions—analyzing the dynamics of market components. Journal of Empirical Finance 4(2), 213–239. Nieto, M. R. and E. Ruiz (2016). Frontiers in var forecasting and backtesting. International Journal of Forecasting 32(2), 475–501. Parkinson, M. (1980). The extreme value method for estimating the variance of the rate of return. Journal of Business, 61–65. Todorov, V. and G. Tauchen (2011). Volatility jumps. Journal of Business & Economic Statistics 29(3), 356–371. Wu, L. (2003). Jumps and dynamic asset allocation. Review of Quantitative Finance and Accounting 20(3), 207–243. 35 