Volatility Determinants: Heterogeneity, Leverage, and Jumps

Transcription

1 Volailiy Deerminans: Heerogeneiy, Leverage, and Jumps Fulvio Corsi Robero Renò February 10, 2009 Absrac We idenify hree main endogenous deerminans in he dynamics of asse price volailiy, namely heerogeneiy, leverage, and jumps. We find ha each of he hree componens plays a significan role in volailiy forecasing and neglecing one of hem is derimenal o he forecasing performance. Imporanly, we find remarkable forecasing power for he negaive pas reurns a all he considered frequencies, which unveils a novel heerogeneous srucure in he leverage effec. We also show, using simulaion sudies, ha he presence of jumps is imporan for wo disinc reasons. Firsly, explicily modeling jumps has rimming effec on he dynamics of he persisen volailiy componen. Secondly, hey have a posiive and significan impac on fuure volailiy, alhough of a shor-lived naure. JEL classificaion: C13; C22; C51; C53 Keywords: Volailiy Forecasing; High Frequency Daa; HAR; Leverage Effec; Jumps. Commens are welcome. We would like o acknowledge Davide Pirino for research assisence and Giampiero Gallo and Roel Oomen for useful suggesions. Universià di Siena, Universiy of Lugano, and Swiss Finance Insiue, fulvio.corsi@lu.unisi.ch Universià di Siena, Diparimeno di Economia Poliica, reno@unisi.i 1

2 1 Inroducion Volailiy forecasing is a key ingredien in many financial problems. However, volailiy dynamics displays well known sylized facs which pose serious challenges o sandard economeric models. Volailiy is a clusered and highly persisen process wih a memory decay of several monhs. Equiy and sock-index volailiies show significan asymmeric response o pas reurns. More precisely, volailiy ends o increase more afer a negaive shock han afer a posiive shock of he same magniude (Bollerslev e al., 2006). This asymmeric reurn-volailiy dependence, firs noed by Black (1976), is usually called he leverage effec. Moreover, price process shows presence of sudden large price changes, he so called jumps, which arguably have an imporan impac on volailiy dynamics. In Corsi (2009) a simple Heerogeneous Auo-Regressive of Realized Volailiy (HAR-RV) model has been proposed o capure he empirical memory persisence of volailiy in a simple and parsimonious way. In his paper, we propose an exended version of he HAR-RV model which considers asymmeric responses of he realized volailiy no only o previous daily reurns bu also o pas weekly and monhly reurns. Our main conribuion is hen o show ha he heerogeneous srucure applies o he leverage effec as well, hus reinforcing he Heerogenous Marke Hyphoesis of Muller e al. (1997). In addiion we sudy he impac on fuure volailiy of jumps measured over he same hree differen horizons. Given he inadequacy of bipower variaion in measuring volailiy in presence of jumps, we use he ess and measures inroduced by Corsi e al. (2008) which provide a beer idenificaion and more precise measuremen of jumps, and uncover he significan impac of jumps on fuure volailiy. We confirm his finding ou-of-sample. We also show, by means of a simulaion sudy, ha he presence of jumps is imporan for wo disinc reasons. Firs, i has a direc impac on volailiy dynamics which may be explained by he presence of conemporaneous jumps in price and volailiy. Second, i has a rimming effec on he volailiy series, which allows a beer fi of he realized volailiy, as suggesed by Andersen e al. (2007). Moreover, we conduc robusness ess o check wheher oher volailiy measures proposed in he lieraure (such as absolue variaion, range, or semivariance) conain addiional useful informaion which are no capured in our model specificaion. The resuls of hese comparisons show ha he oher volailiy measures eiher drop ou or only marginally conribue o he performance of he model. Hence, he proposed model seems o capure he main deerminans of volailiy dynamics. Summarizing, in his paper we are proposing a relaively simple model ha incorporaes hree main deerminans of volailiy dynamics, namely: heerogeneiy, leverage and jumps. When esimaing his model on he S&P500 ime series, we find ha each componen has a disinc forecasing power, boh in-sample and ou-of sample. While, oher daa-driven measures of volailiy only marginally add o our model. The paper is organized as follows. Secion 2 reviews he HAR model and presens is possible exensions 2

3 wih heerogeneous leverage and jumps. Secion 3 describes he empirical in-sample and ou-of-sample analysis on a long serie of high frequency S&P500 fuures daa. Secion 4 discuss he resuls in he ligh of wo Mone Carlo simulaion models wih independen and conemporaneous jumps and Secion 5 conains some concluding remarks. 2 Modelling volailiy Assume ha he sae variable X, which may be hough as an economic variable (an ineres rae or a sock log-price), is driven by he sochasic process: dx = µ d + σ dw + c dn (2.1) where µ is predicable, σ is cádlág and N is a doubly sochasic Poisson process whose inensiy is an adaped sochasic process λ, he imes of he corresponding jumps are (τ j ) j=1,...,nt and c j are i.i.d. adaped random variables measuring he size of he jump a ime τ j. In pracice, e.g. for risk managemen purposes, we are ineresed in forecasing he quadraic variaion defined as: σ = +1 σ 2 sds + c 2 τ j, τ j +1 where he ime uni is one day. We esimae quadraic variaion using n observaions of he sae variable in he ineval [0, T]. The mos popular esimaor is realized volailiy, which, afer defining,j X = X +j/n X +(j+1)/n, is given by: n 1 RV = (,j X) 2 (2.2) j=0 which is a consisen esimaor, as n, of σ, see Andersen e al. (2003) for a review. Oher esimaors have been devised, such as he range (see e.g. Alizadeh e al and Brand and Jones 2006) or refinemen of realized volailiy o accoun for he presence of microsrucure noise, as hose in Zhang e al. (2005), Barndorff-Nielsen e al. (2008), or Jacod e al. (2007). We indicae by V a generic unbiased esimaor of quadraic variaion such ha (working wih logarihms o avoid negaiviy issues): log σ = log V + ω (2.3) where ω is zero mean and finie variance measuremen error. 1 We are ineresed in modelling he dynamics of σ, ha is he dynamics of quadraic variaion, a opic which has received growing aenion in he las decade. 1 In our empirical analysis, we use he wo-scales esimaor of Zhang e al. (2005). 3

4 2.1 Heerogeneiy The need for heerogeneiy of volailiy componens, advocaed by Muller e al. (1997), has been reconsidered in he work of Corsi (2009) by making use of he concep of volailiy cascades. In wha follows, we review his laer approach working wih logarihmic ransformaions o avoid negaiviy issues and ge approximaely Normal disribuion for he volailiy measure. Consider he aggregaed values of log V, defined as: (n) log V = 1 (log n V log V ) n+1 and assume having wo differen ime scales, of lengh n 1 and n 2, wih n 1 > n 2. For he larges ime scale, assume ha σ, once aggregaed as in (2.4) is deermined by: log σ (n1) +n 1 = c (n1) + β (n1) log where ε (n1) is IID zero mean and finie variance noise independen on ω. (2.4) (n1) V +ε (n1) +n 1 (2.5) I has been recenly suggesed ha volailiy over longer ime inervals has sronger influence on hose over shorer ime inervals han conversely, suggesing a volailiy cascade from low o high frequencies. 2 This can be economically explained by noicing ha for shor-erm raders he level of long erm volailiy maers because i deermines he expeced fuure size of rends and risk. On he oher hand, he level of shor-erm volailiy does no affec he rading sraegies of long-erm raders. The shorer ime scale (n 2 ) is assumed o be influenced by he expeced fuure value of he larges ime scale (n 1 ), so ha: [ ] log σ (n2) +n 2 = c (n2) + β (n2) (n2) log V +n 2 +δ (n2) E log σ (n1) +n 1 + ε (n2) +n 2 ; (2.6) wih ε (n2) IID zero mean and finie variance noise independen on ε (n1) and ω. By subsiuion, and using equaion (2.3), his gives: (n2) log V +n 2 = c + β (n2) log where ε is IID noise depending on ε (n1) (n2) V +β (n1) (n1) log V +ε ; (2.7), ε (n2), ω. The model (2.7) can be easily exended o d horizons of lengh n 1 > n 2 >... > n d. Typically, hree componens are used wih lengh n 1 = 22 (monhly), n 2 = 5 (weekly), n 3 = 1 (daily). Since volailiy a shorer ime horizons is influenced by volailiy a longer horizons, he auo-correlaion funcion of he model and hence is memory persisence increases. Thus, even if he HAR model does no formally belong o he class of long memory processes, i fis he persisence properies of financial daa as well as rue long memory models, such as he fracionally inegraed one, which, however, are much more complicaed o esimae and o deal wih (see he review of Banerjee and Urga 2005). The HAR model has been employed in several applicaions in he lieraure. Corsi e al. (2008) use i o sudy he volailiy of realized volailiy; Ghysels e al. (2006) and Forsberg and Ghysels (2007) compare 2 See Müller e al. (1997), Arneodo e al. (1998), Lynch and Zumbach (2003). However, he HAR model would hold even if we allow he shor-erm volailiy o affec he long-erm volailiy, alhough his would be a odds wih he empirical findings (see Secion 3.1). 4

5 his model wih he MIDAS model; Andersen e al. (2007) use an exension of his model o forecas he volailiy of sock prices, foreign exchange raes and bond prices; Clemens e al. (2008) implemen i for risk managemen wih VaR measures; Bollerslev e al. (2008) use i o analyze he risk-reurn radeoff. 2.2 Leverage effecs I is well known ha equiies and sock indexes ofen exhibi he so called leverage effec, i.e. volailiy ends o increase more afer a negaive shock han afer a posiive shock of he same size. By exending he Heerogeneous Marke Hypohesis approach o leverage effec, we consider asymmeric responses of realized volailiy no only o previous daily reurns bu also o pas weekly and monhly reurns. We model such heerogeneous leverage effecs by inroducing asymmeric reurn-volailiy dependence a each level of he cascade considered in he above secion. Define daily reurns r = X X 1 and pas aggregaed negaive and posiive reurns as: r (n)+ = 1 n (r r n ) I {(r+...+r n) 0} (2.8) r (n) = 1 n (r r n ) I {(r+...+r n)<0} (2.9) where I { } denoes he indicaor funcion. We assume ha inegraed volailiy is deermined by he cascade: log σ (n1) +n 1 = c (n1) + β (n1) log log σ (n2) +n 2 = c (n2) + β (n2) log which now gives: (n2) log V +n 2 = c + β (n2) log + γ (n2) r (n2) (n1) V +γ (n1) r (n1) + γ (n1)+ r (n1)+ + ε (n1) +n 1 [ ] (n1) V +γ (n2) r (n2) + γ (n2)+ r (n2)+ + δ (n2) E log σ (n1) +n 1 + ε (n2) +n 2 (n2) V +β (n1) (n1) log V (2.10) + γ (n2)+ r (n2)+ + γ (n1) r (n1) + γ (n1)+ r (n1)+ + ε We hen posulae ha leverage effecs influence each marke componen separaely, and ha hey appear aggregaed a differen horizons in he volailiy dynamics. Noe ha he inclusion of boh negaive and posiive reurns is equivalen, by lineariy, o he inclusion of negaive (or posiive) and oal reurns. 2.3 Jumps The imporance of jumps in financial economerics is rapidly growing. Recen research focusing on jumps deecion and volailiy measuring in presence of jumps includes Barndorff-Nielsen and Shephard (2004); Mancini (2007), Lee and Mykland (2007), Jiang and Oomen (2008), Aï-Sahalia and Mancini (2008), Aï-Sahalia and Jacod (2008), Chrisensen e al. (2008), and Boud e al. (2008). Andersen e al. (2007) suggesed ha he coninuous volailiy and jump componen have differen dynamics and should hus 5

6 be modelled separaely. In his secion, we follow closely Corsi e al. (2008) using he C-Tz saisics o deec he occurrence of he jump in a single day, and hreshold bipower variaion o measure he coninuous par of inegraed volailiy, defined as: TBPV = 2 n 2,j X,j+1 X I π {,jx 2 ϑ I j 1} {,j+1x 2 ϑ j} (2.11) j=0 where ϑ is a hreshold funcion which we esimae as in Corsi e al. (2008). This coninuous volailiy esimaor has beer finie sample properies han sandard bipower variaion and provides more accurae jump ess, which allows for a correced separaion of coninuous and jump componens. To his purpose, we fix a confidence level α and esimae he jump componen as: ) + J = I {C-Tz>Φα} ( V TBPV (2.12) where Φ α is he value of he sandard Normal disribuion corresponding o he confidence level α, and x + = max(x,0). The corresponding coninuous componen is defined as: C = V J, (2.13) which is equal o V if here are no jumps in he rajecory and o TBPV if a jump is deeced wih he C-Tz saisics. Figure 1 repors he percenage conribuion of jumps esimaed by (2.12) o oal quadraic variaion compued on a 3-monh and 1-year moving window for he full S&P 500 fuures sample. In line wih he resuls in Andersen e al. (2007) and Huang and Tauchen (2005) we find a jumps conribuion varying beween 2% and 20% of oal variaion (wih an overall sample mean of abou 6%), showing a higher level a he beginning of he sample and an increasing rend oward he end of he sample period. In he volailiy cascade we assume ha C and J ener separaely a each level of he cascade, ha is: log σ (n1) +n 1 = c (n1) + α (n1) log(j (n1) log σ (n2) +n 2 = c (n2) + α (n2) log(j (n2) +1) + β (n2) log C (n2) [ ] + δ (n2) E σ (n1) +1 + ε (n2) +n 2 originaing he model: (n2) log V +n 2 = c + α (n1) log(j (n1) + γ (n2) r (n2) +1) + β (n1) log C (n1) +γ (n1) r (n) + γ (n2)+ r (n)+ + ε (n1) +n 1 +1) + α (n2) log(j (n2) + γ (n2)+ r (n2)+ +γ (n2) r (n) + γ (n2)+ r (n)+ +1) + β (n2) log C (n2) +β (n1) log C (n1) (2.14) + γ (n1) r (n1) + γ (n1)+ r (n1)+ + ε 2.4 The LHAR-CJ model Combining heerogeneiy in realized volailiy, leverage, and jumps we consruc he Leverage Heerogeneous Auo-Regressive wih Coninuous volailiy and Jumps (LHAR-CJ) model. As i is common in 6

7 Jump Conribuion o Toal Variaion monh window 1 year window Figure 1: Percenage conribuion of daily jump esimaed by (2.12) o oal quadraic variaion measured over a moving window of 3-monh (doed line) and 1-year (solid line) for he S&P500 fuures from January 1990 o December 2007 (4344 days). The C-Tz saisics in (2.12) is compued wih a confidence inerval α = 99.9%. pracice, we use hree componens: daily, weekly and monhly. Hence, he proposed model reads: (h) log V +h = c + α (d) log(1 + J ) + α (w) log(1 + J (5) ) + α (m) log(1 + J (22) ) (2.15) + β (d) log C + β (w) log C (5) + β (m) log C (22) + γ (d)+ r + + γ (w)+ r (5)+ + γ (m)+ r (22)+ + γ (d) r + γ (w) r (5) + γ (m) r (22) + ε (h), We esimae model (2.15) by OLS wih Newey-Wes covariance correcion for serial correlaion. In order o make muliperiod predicions we will esimae he model considering he aggregaed dependen variable (h) log V +h wih h ranging from 1 o 22 i.e. from one day o one monh. While, sricly speaking, models wih h > 1 would require a cascade specificaion wih longer frequencies muliple of h, for simpliciy and comparison purposes, we will always reain he sandard cascade specificaion wih he hree naural frequencies of one day, one week and one monh. This can be viewed as a simplifying approximaion jusified by is empirically good performances. 3 Empirical evidence The purpose of his secion is o empirically analyze he main deerminans of fuure asse volailiy. Our daa se covers a long ime span of almos 18 years of high frequency daa for he S&P 500 fuures from he 7

8 January 1990 o June 2007 (4,344 days). In order o miigae he impac of microsrucure effecs on our esimaes, he daily volailiies V are compued wih he wo-scales esimaor proposed by Zhang e al. (2005). 3 Aï-Sahalia and Mancini (2008) show ha using he wo-scales esimaor insead of sandard realized volailiy measures yields significan gains in volailiy forecasing. The TBPV measure (2.11) for jump deecion is compued a he sampling frequency of 5 minues (corresponding o 84 reurns per day). 3.1 In-sample analysis The LHAR-CJ model is esimaed using, as a dependen variable, realized volailiy aggregaed a differen horizons. The resuls of he esimaion of he LHAR-CJ when forecasing he S&P500 realized volailiy a 1 day, 1 week, 2 weeks and 1 monh are repored in Table 1, ogeher wih heir saisical significance evaluaed wih he Newey-Wes robus -saisic. As usual, all he coefficiens of he hree coninuous volailiy componens are posiive and, in general, highly significan. I is, however, ineresing o remark ha, while he coefficien which measures he impac of monhly volailiy on daily volailiy is highly significan a all horizons, he opposie does no hold, confirming he hierarchical asymmeric propagaion of he volailiy cascade presened in Secion 2. The daily and weekly jump componens remain highly significan and posiive for he shorer horizon realized volailiy and become insignifican for he longer ones. In paricular, while he daily jumps say srongly significan up o he weekly horizon, he weekly jumps remain significan up o he 2-week horizon. Figure 2 plos he -saisics of he impac of he daily jump on aggregaed volailiy a differen ime horizons, confirming, wih is rapid decline, ha daily jumps affecs fuure volailiies only over a shor period of abou one week. The jumps aggregaed a monhly level urn ou o be always insignifican and migh be removed by our specificaion. The mos ineresing resul is, however, he srong significance (wih an economically sound negaive sign) of he negaive reurns a all he daily, weekly and monhly frequencies which unveils an heerogeneous srucure in he leverage effec as well. No only daily negaive reurns affec he nex day volailiy (he well know leverage effec), bu, in addiion, also he negaive reurns of he pas week and pas monh have an impac on forhcoming volailiy, which is even sronger han ha of he previous day. This finding leads o he conclusion ha he marke aggregaes daily, weekly and monhly memory, observing and reacing o price declines happened in he pas week and monh. To our knowledge, his is a novel empirical finding ha furher confirms he views of he Heerogeneous Marke Hypohesis. On he conrary, posiive reurns are almos insignifican in predicing fuure volailiy and may be removed from a baseline model specificaion. However, i is ineresing o remark ha he -saisics 3 The wo-scales esimaor combines wo realized volailiies compued a wo differen frequencies, where he slower one is compued by subsampling and averaging while he faser one (being a proxy for he variance of microsrucure noise) is used for bias correcion. In our implemenaion of he wo-scales esimaor we use a slower frequency of en icks reurns. 8

9 S&P500 LHAR in-sample regression Variable One day One week Two weeks One monh c 0.571* 0.933* 1.326* 2.031* (6.993) (3.263) (2.751) (2.625) C 0.236* 0.145* 0.101* (10.205) (7.354) (4.299) (1.531) C (5) 0.372* 0.343* 0.289* 0.177* (11.450) (7.961) (5.652) (2.760) C (22) 0.264* 0.310* 0.326* 0.340* (9.652) (6.490) (4.863) (3.507) J 0.038* 0.016* (5.004) (3.213) (1.444) (0.119) J (5) 0.012* 0.013* 0.014* (3.031) (2.390) (2.499) (1.274) J (22) (-0.323) (-0.711) (-1.251) (-1.113) r * * * * (-8.074) (-7.874) (-5.481) (-3.401) r (5) * * * (-4.629) (-2.013) (-2.108) (-1.920) r (22) * * * * (-2.209) (-2.485) (-2.205) (-2.256) r (0.102) (0.920) (1.523) (1.552) r (5) (-1.157) (-0.232) (0.440) (0.824) r (22) * 0.021* 0.028* (1.496) (2.319) (2.697) (2.859) R RMSE Table 1: OLS esimae for LHAR-CJ regressions, model (2.15), for S&P500 fuures from January 1990 o December 2007 (4, 344 observaions). The LHAR-CJ model is esimaed on 1-day, 1-week, 2-week and 1-monh realized volailiy. The significan jump are compued using a criical value of α = 99.9%. Repored in parenhesis are -saisics based on Newey-Wes correcion. 9

10 of monhly posiive reurns increase wih he forecasing horizon, becoming mildly significan a weekly and longer horizons. This suggess ha, for longer horizons, price rends affec fuure volailiy, mildly corroboraing he resuls in Zumbach (2005). In order o evaluae he relaive conribuion of he differen volailiy deerminans we compare he in-sample predicion of he LHAR-CJ model over differen horizons wih hose of he sandard HAR model and he HAR model wih jumps bu wih no leverage effecs (he HAR-CJ model of Corsi, Pirino, and Renò 2008). For each horizon, ranging from one day o one monh, he forecass are obained by firs esimaing he parameers of he models on he full sample and hen performing a series of saic one-sep-ahead forecass. The forecass of he differen models are evaluaed on he basis of he RMSE in predicing he square roo of V. The resuls, repored in Figures 3, show unambiguously ha he inclusion of boh he heerogeneous jumps and he heerogeneous leverage effecs considerably improve he forecasing performance of he S&P 500 volailiy a any forecasing horizon. In paricular, he inclusion of heerogeneous leverage effec provides a relevan overall benefi in he in-sample performance. 3.2 Robusness o oher volailiy measures In he lieraure many volailiy measures have been proposed o beer capure he dynamics of volailiy. Forsberg and Ghysels (2007) proposed he use of realized absolue variaion (RAV) which shows a more persisen dynamics han realized volailiy being more robus o microsrucure noise and jumps. The range has also been found o be significan by many auhors, see e.g. Brand and Jones (2006) and Engle and Gallo (2006). Moivaed by he analysis of Bandi e al. (2008) who found liquidiy o be a significan facor in asse pricing, we also compue he sum of squared ick-by-ick reurns as a liquidiy measure and employ i as a volailiy facor. Recenly, Barndorff-Nielsen e al. (2008) proposed he realized semivariance as he sum of square negaive reurns o capure he impac on volailiy of downward price pressures. Visser (2008) combines RAV and semivariance by aking he sum of negaive absolue squared reurns. In he spiri of Forsberg and Ghysels (2007), we compare he relaive explanaory power of differen volailiy measures by esimaing he following se of models (for space concerns we limi ourself o he one day horizon, hus we do no include posiive reurns which are significan a longer horizons only). Firs, we esimae a baseline LHAR-CJ model, conaining as explanaory variables: heerogeneous coninuous volailiy, heerogeneous negaive reurns and daily jumps. Then we add he differen volailiy measures o he baseline LHAR-CJ model. Some of hese measures urn ou o be fairly relaed o he jump one (range and semivariance). For hose measures we also esimae models where he daily jump regressor is removed so ha a direc performance comparison wih he LHAR-CJ is possible. Esimaion resuls are repored in Table 2. The liquidiy proxy (LQ) and he realized absolue variaion (RAV) compued a 5-minue frequency urn ou o be no significan when included in he LHAR-CJ model. In paricular, our resul for he 10

11 6 -saisics of daily jump coefficiens 4 sa h Figure 2: -saisics of daily jump coefficiens for LHAR-CJ model esimaed on S&P500 from January 1990 o December 2007 (4344 days) as a funcion of he forecasing horizon h. In-sample RMSE HAR HAR CJ LHAR CJ Figure 3: RMSE of saic in sample one-sep ahead forecass for realized volailiy ranging from 1 day o 1 monh of he S&P500 from January 1990 o December 2007 (4344 observaions). The forecasing models are he sandard HAR wih only heerogeneous volailiy, he HAR-CJ wih heerogeneous jumps and he LHAR-CJ model. 11

12 RAV seems o conras wih findings in he lieraure of a higher explanaory power of RAV vs. RV due o a higher robusness of he RAV o microsrucure noise and jumps. However, here he 5-minue RAV is confroned wih he ick-by-ick wo-scales measures cleaned from he jumps componen by he C-Tz es, hence wih a highly precise measure which (conrary o RV) is also robus o microsrucure noise and jumps. In line wih previous lieraure, we find ha he range has a significan impac on fuure volailiy. However, i seems o be mainly a subsiue for coninuous volailiy and jumps, which is no oally surprising since he range is an esimaor (hough noisy) of oal quadraic variaion. Indeed, when he range replaces he jumps (LHAR-Range model), he coefficiens of daily coninuous volailiy almos halves. The R 2 of he wo compeing regressions (LHAR-Range and LHAR-CJ) is pracically he same. When he range is insered ogeher wih he jumps (LHAR-CJ-Range), boh he coefficiens of daily volailiy and jumps decrease, alhough hey remain highly significan. While, he significance of he heerogeneous leverage effec is unouched by he presence of he range. The R 2 of he encompassing regression increases marginally. We hus conclude ha he range, while parially proxying for boh volailiy and jump, is also able o capure some oher effec which is no capured by oher variables, bu i adds very lile o he economic and saisical value of he LHAR-CJ model. We found similar resuls for he realized semivariance of Barndorff-Nielsen e al. (2008) and he downward absolue power variaion of Visser (2008) (being very similar, only he resul for he realized semivariance are repored). 4 As for he range, realized semivariance (LHAR-semiRV and LHAR-CJ-semiRV) is significan in explaining fuure volailiy, and i seems very correlaed wih boh he daily wo-scales esimaor and he jumps (ypically depleing he significance of he corresponding coefficiens wihou oally removing i), while unrelaed wih he leverage. The conribuion of he realized semivariance o he model performance is very similar o ha of he range. However, when hey are included ogeher in he LHAR-CJ model (LHAR-CJ-Range-semiRV) hey boh remain significan. Summarizing, he resuls of his secion show ha when he oher volailiy measures proposed in he lieraure are insered in he baseline LHAR-CJ model hey eiher drop ou or only marginally conribue o he performance of he model. Hence, he LHAR-CJ model seems o capure he main deerminans of volailiy dynamics. 3.3 Ou-of-sample analysis In his secion, we appreciae he performance of he LHAR-CJ model on he basis of rue ou-of-sample analysis. For he ou-of-sample forecas of V on he [,+h] inerval we keep he same forecasing horizons ranging from one day o one monh and reesimae he model a each day on an increasing window of all he observaion available up o ime 1. The ou of sample forecasing performance for he square roo 4 All resuls are available from he auhors upon reques. 12

13 Ou-of-sample RMSE HAR HAR CJ LHAR CJ Figure 4: RMSE of ou-of-sample forecass for realized volailiy ranging from 1 day o 1 monh of he S&P500 from January 1994 o December 2007 (3344 observaions, he firs 1000 observaion are used o iniialize he models). The forecasing models are he sandard HAR wih only heerogeneous volailiy, he HAR-CJ wih heerogeneous jumps and he LHAR-CJ model. of V in erms of RMSE is repored in Figure 4. The superioriy of he LHAR-CJ model a all horizons is confirmed, validaing he imporance of including boh he heerogeneous leverage effecs and jumps in he forecasing model. I is ineresing o noe ha he improvemens in forecasing performance due o he inclusion of jumps (HAR-CJ vs HAR) is sill valid ou-of-sample, reinforcing he resuls in Corsi e al. (2008). For longer horizons he performance differences among he hree models end o decrease. Finally, i is imporan o noe ha he inclusion of he jump componen helps also in forecasing longer horizon volailiy, which seems a odds wih he shor-lived naure of he impac of jumps on fuure volailiy, as winessed by he significance of he α coefficiens in Table 1. To clarify his issue, we perform a Mone Carlo simulaion analysis described in he following secion. 4 A simulaion sudy We evaluae our empirical resuls for jumps hrough he lens of a Mone Carlo simulaions. We simulae he sock index price wih he flexible specificaion of Eraker e al. (2003), ha is: dy µ = d + V 1 0 dw + ξy dn y dv κ(θ V ) σ v ρ σ v 1 ρ2 σ v ξ v dn V (4.1) where W is a bidimensional Brownion moion and dn y and dn V are Poisson processes wih inensiy 13

14 λ y and λ V respecively; ξ y is normally disribued, while ξ V has an exponenial law. As in Eraker e al. (2003), we consider wo cases: he case in which dn y is independen from dn V (wha hey name he SVIJ model) and he case in which dn y = dn V (wha hey name he SVCJ model), and we hold heir erminology. We use exacly he parameers esimaed by Eraker e al. (2003) for he S&P500 ime series. Figure 5 and 6 repor he resuls. Explicily including he jump componen has a direc benefi for boh he SVIJ (independen jumps) and he SVCJ (conemporaneous jumps) specificaion. In he SVIJ case, jumps has no impac on fuure volailiy, bu here is sill a benefi in removing he jump componen. Indeed, in his model he persisence is conveyed only by he coninuous volailiy, while oal quadraic variaion (which is esimaed by realized volailiy) also conain he memoryless jumps. Thus, by separaing he jumps from he persisen par in he explanaory variables, a beer model specificaion is obained. We conclude ha, when he memory of volailiy is mainly conained in he coninuous par of quadraic variaion, here is sill a poenial benefi in removing jumps even if hey do no impac on fuure volailiy. This benefi persis also for long horizon forecass. Imporanly, in his case, he jump componen is found o be insignifican. In he SVCJ he performance improvemens of he models wih jumps is also given by he direc link beween jumps in price and volailiy, which explains he overall lower RMSE (also for he HAR model, which incorporaes jumps direcly in quadraic variaion esimaed via realized volailiy). When a jump occurs in price, i also occurs in volailiy and i is posiive. Thus, when here is a jump in price, volailiy becomes higher and i says higher because of is memory persisence. Tha is why jumps are (conrary o he SVIJ case) found o be significan in explaining fuure volailiy in SVCJ models. Hence, our simulaion resuls show ha a possible mechanism explaining he significan impac of jumps on fuure volailiy is given by conemporaneous jumps in price and volailiy. Moreover, i confirms ha he significance of he jumps coefficiens on our empirical analysis provides an indicaion of he presence of a genuine forecasing power of jumps on fuure volailiy. Hence, he similariy beween he figures reporing he Newey-Wes correced -saisics of he daily jump coefficien esimaed on he simulaed SVCJ model (Figure 6 righ panel) and on he empirical S&P500 (Figure 2), confirms he presence of a genuine forecasing power of jumps on fuure S&P volailiy. On he oher hand, he heerogeneous leverage effec found in real daa canno be compleely explained by model 4.1. Indeed, he presence of a negaive coefficien ρ 0.5 (esimaed on S&P 500 daa) is able o explain only shor-period leverage effec, by propagaing negaive reurns ino conemporaneous, and by memory persisence, fuure volailiy. While, in he real daa, we provided evidence for srong heerogeneous leverage effec, being also he weekly and monhly negaive componens highly significan. The model specificaion 4.1 is hen insufficien o explain our resuls which demand for a more complicaed coninuous process wih a richer specificaion. 14

15 SVIJ SVCJ HAR HAR CJ LHAR CJ HAR HAR CJ LHAR CJ Figure 5: RMSE of realized volailiy forecas ranging from 1 day o 1 monh of 4000 days simulaed daa from SVIJ (lef) and SVCJ(righ) model. The forecasing models are he sandard HAR wih only heerogeneous volailiy, he HAR-CJ wih heerogeneous jumps and he LHAR-CJ model. SVIJ SVCJ sa 2 sa h h Figure 6: -saisics of daily jump coefficiens for LHAR-CJ model esimaed on 4000 simulaed daily daa from SVIJ (lef) and SVCJ (righ) model as a funcion of he forecasing horizon h. 5 Conclusions This paper presens a new model for volailiy forecasing which isolaes hree main deerminans of volailiy dynamics, namely heerogeneous pas volailiy, heerogeneous pas negaive reurns and jumps. We find ha each componen plays a differen role a differen forecasing horizons, bu all he hree are highly significan and neglecing each one of hem is derimenal o he forecasing performance of he model. Moreover, when oher volailiy measures proposed in he lieraure are insered in he LHAR-CJ model hey eiher drop ou or only marginally conribue o he performance of he model confirming he abiliy of he LHAR-CJ model o capure he main deerminans of volailiy dynamics. 15

16 Explicily modelling he jump componen is imporan for wo disinc reasons. Firs, i has a rimming effec on he dynamics of he persisen componen of volailiy which allows a beer predicion of fuure volailiy, confirming Andersen e al. (2007). Secondly, as suggesed in Corsi e al. (2008), hey have a direc posiive and significan impac on fuure volailiy. Moreover, here are evidences ha his direc impac of jumps is of a shor lived naure. If, as i seems reasonable, volailiy is a measure of he uncerainy of he marke abou is fundamenal values, our findings can be inerpreed as follows: afer a jump (usually a marke crash) he marke akes a longer period o reassess is fundamenal value by dissipaing he uncerainy creaed by he jumps; during his period residual uncerainy generae higher volailiy. Our simulaed experimens indicaed ha his mechanism can be saisically reproduced by a model having conemporaneous jumps in price and volailiy. On he oher hand, while he mechanism of leverage effecs on volailiy dynamics is sill no well undersood, we find ha no only daily bu also weekly and monhly negaive pas reurns are highly significan and have a remarkable forecasing power on fuure volailiy. This novel effec seems o confirm he heerogeneous srucure of he marke and canno be explained by coninuous-ime models, hough flexible, as he ones specified so far in he lieraure. We also find ha, a longer horizons, posiive reurns (price rends) have an impac on fuure volailiy. We conclude by noing ha our model is very simple o implemen, as i does no requires sofisicaed compuaional echnique. The esimaion of he model parameers can be performed hrough a simple OLS regression, and he compuaion of he explanaory variables is rivial. We hink ha, for all he aforemenioned reasons, his model may be effecively used for risk managemen. 16

17 Table 2: Esimaed parameers, RMSE, and R 2 of alernaive specificaions of he baseline LHAR-CJ model; -saisics are in parenhesis. Variable LHAR-CJ LHAR-CJ-LQ LHAR-CJ-RAV LHAR-Range LHAR-CJ-Range LHAR-SemiRV LHAR-CJ-SemiRV LHAR-CJ- Range-SemiRV 17 cons 0.586* 0.606* * 0.592* 0.709* 0.662* 0.669* (7.735) (7.044) (1.113) (8.120) (7.829) (9.142) (8.281) (8.406) C 0.246* 0.265* 0.184* 0.124* 0.181* 0.083* 0.146* 0.077* (11.026) (8.767) (2.462) (5.675) (7.281) (3.426) (4.782) (2.351) C (5) 0.352* 0.355* 0.352* 0.381* 0.360* 0.370* 0.356* 0.364* (11.445) (11.491) (11.434) (12.391) (11.804) (11.894) (11.602) (11.959) C (22) 0.273* 0.278* 0.270* 0.283* 0.274* 0.277* 0.271* 0.271* (10.257) (10.364) (10.397) (10.398) (10.312) (10.196) (10.183) (10.221) r * * * * * * * * (-8.721) (-8.798) (-8.748) (-8.007) (-7.818) (-6.248) (-6.428) (-5.519) r (5) * * * * * * * * (-5.349) (-5.330) (-5.102) (-5.996) (-5.504) (-5.410) (-5.187) (-5.335) r (22) * * * * * * (-2.012) (-2.037) (-1.955) (-1.981) (-2.010) (-2.198) (-2.157) (-2.156) J 0.048* 0.052* 0.039* 0.036* 0.030* (6.872) (6.626) (3.243) (4.870) (3.650) (1.927) LQ (-0.850) RAV (0.833) Range 0.082* 0.059* 0.060* (7.963) (5.521) (5.662) SemiRV 0.142* 0.101* 0.104* (7.806) (4.746) (4.944) RMSE R

18 SVIJ regression SVCJ regression 1 day 1 week 2 weeks 1 monh c 0.350* 0.839* 1.327* 2.245* (2.931) (2.357) (2.465) (2.974) C 0.630* 0.564* 0.499* 0.401* (15.576) (11.825) (8.817) (6.041) C (5) 0.345* 0.356* 0.369* 0.377* (7.905) (7.551) (6.606) (5.683) C (22) (-1.752) (-1.313) (-1.315) (-1.592) J (-1.493) (-1.141) (-0.819) (-1.156) r * * * * (-5.590) (-4.355) (-3.488) (-3.209) r (5) (-1.068) (-1.806) (-1.507) (-0.904) r (22) (1.564) (0.982) (0.231) (-0.244) 1 day 1 week 2 weeks 1 monh c 0.419* 1.049* 1.690* 2.849* (2.905) (2.248) (2.397) (3.042) C 0.464* 0.425* 0.375* 0.312* (11.514) (8.151) (5.875) (4.216) C (5) 0.531* 0.546* 0.568* 0.550* (13.154) (10.320) (9.094) (7.681) C (22) * * * * (-2.613) (-2.179) (-2.279) (-2.584) J 0.062* 0.059* 0.054* 0.046* (4.291) (3.664) (2.809) (2.005) r * * * * (-3.848) (-3.377) (-2.690) (-2.477) r (5) * * * (-2.433) (-1.837) (-2.454) (-2.595) r (22) (0.787) (-0.882) (-1.590) (-1.911) Table 3: OLS esimae for baseline LHAR-CJ model, for S&P500 fuures from January 1990 o December 2007 (4344 observaions). The LHAR-CJ model is esimaed on 1-day, 1-week, 2-week and 1-monh realized volailiy. The significan jumps are compued using a criical value of α = 99.9%. Repored in parenhesis are -saisics based on Newey-Wes correcion. 18

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