Predictable Dynamics in Higher Order Risk-Neutral Moments: Evidence from the S&P 500 Options *

Transcription

1 Predicable Dynamics in Higher Order Risk-Neural Momens: Evidence from he S&P 500 Opions * Michael Neumann a and George Skiadopoulos b Journal of Financial and Quaniaive Analysis, forhcoming Absrac We invesigae wheher here are predicable paerns in he dynamics of higher order risk-neural momens exraced from he marke prices of S&P 500 index opions. To his end, we conduc a horse race among alernaive forecasing models wihin an ou-of-sample conex over various forecasing horizons. We consider boh a saisical and an economic seing. We find ha higher risk-neural momens can be saisically forecased. However, only he one-day-ahead skewness forecass can be economically exploied. This economic significance vanishes once we incorporae ransacion coss. The resuls have implicaions for he dynamics of implied volailiy surfaces. Keywords: Implied volailiy surface, Marke efficiency, Model confidence se, Opion sraegies, Risk-neural skewness, Risk-neural kurosis. JEL Classificaion: C53, C58, G10, G13, G17. * We would like o hank Paul Malaesa (he edior) and an anonymous referee for heir consrucive, simulaing, and horough commens. We would also like o hank Turan Bali, Alejandro Bernales, Tim Bollerslev, Rober Engle, Daniel Giamouridis, Eric Ghysels, Massimo Guidolin, Erik Hjalmarsson, George Kapeanios, Eirini Konsaninidi, Alexandros Kosakis, Marc Paolella, Paul Schneider, Efhymios Tsionas, Greg Vilkov, and paricipans a he 2011 Annual Conference of he Sociey for Financial Economerics (Chicago), 2011 Annual Conference on Research in Economic Theory and Economerics (Milos), and he 2011 Annual Meeing of he German Finance Associaion (Regensburg) for useful discussions and commens. The firs auhor acknowledges financial suppor from he Sociey for Financial Economerics and he second auhor from he Research Cenre of he Universiy of Piraeus and he Fundação Ciência e Tecnologia's gran number PTDC/EGE-ECO/099255/2008. Any remaining errors are our responsibiliy. a School of Economics and Finance, Queen Mary, Universiy of London, m.neumann@qmul.ac.uk, Posal Address: 327 Mile End Road, London E1 4NS, Unied Kingdom. Tel: b Corresponding auhor. Deparmen of Banking and Financial Managemen, Universiy of Piraeus, Greece, Financial Opions Research Cenre, Warwick Business School, Universiy of Warwick, and Cass Business School, Ciy Universiy, UK, gskiado@unipi.gr, Posal Address: Universiy of Piraeus, Deparmen of Banking and Financial Managemen, Karaoli and Dimiriou 80, Piraeus 18534, Greece, Tel:

2 I. Inroducion A any given poin in ime, he cross-secion of marke opion prices across srike prices for a given expiry deermines he momens of he risk-neural probabiliy densiy funcion (ermed risk-neural momens, RNMs hereafer) of he underlying asse as well as he implied volailiy surface (IVS). An IVS maps he implied volailiies exraced from marke opion prices o differen srike prices and mauriies. Ineresingly, he evoluion of RNMs reflecs ha of IVS and hence ha of marke opion prices. This is because changes in risk-neural volailiy, skewness, and kurosis are relaed o he level, slope, and curvaure of he IVS, respecively (Zhang and Xiang (2008)). Undersanding he dynamics of he higher order RNMs is hus of imporance o boh academics and praciioners. This is because i sheds ligh on he design of opion pricing models ha no only price and hedge opions accuraely a a given poin in ime bu also a differen poins in ime. This is a prerequisie for a model o be credible (for he same raionale, see Bakshi, Cao, and Chen (1997), and Dumas, Fleming, and Whaley (1998)). Furhermore, from a rader s perspecive, forecasing he changes in RNMs can help developing profiable risk-neural volailiy, skewness, and kurosis rading sraegies. We underake a comprehensive sudy of he dynamics of higher RNMs exraced from he prices of he S&P 500 index opions, and in paricular invesigae wheher hese can be used o forecas heir subsequen changes. 1 To his end, we conduc a horse race among a number of models, and evaluae heir ou-of-sample forecasing performance under a saisical and economic seing. In paricular, we apply formal saisical ess o assess wheher here are predicable paerns in he dynamics of RNMs, and deermine he se of bes performing models according o he model confidence se mehodology proposed by Hansen, Lunde, and Nason (2011). Nex, we assess he economic significance of he forecass of he RNMs. We propose and apply risk-neural skewness and kurosis opion rading sraegies based on he forecased RNMs and evaluae heir performance. 1 This quesion is disinc from he quesion of wheher RNMs can forecas heir realized counerpars as well as oher financial and economic variables (see Xing, Zhang, and Zhao (2010), Bakshi, Panayoov, and Skoulakis (2011), Kosakis, Panigirzoglou, and Skiadopoulos (2011)). 2

3 To address he posed quesions, we employ a daase of European index opions wrien on he S&P 500 spanning he period This is a rich period ha includes bull and bear regimes, as well as he recen subprime crisis period; is use is necessary o uncover he dynamics of RNMs given ha hese depend on relaively rare evens, oo. To check he robusness of our resuls, we examine higher RNMs of alernaive consan horizons and forecas hem over differen forecasing horizons (daily, weekly, and monhly). We find ha he higher RNMs can be saisically forecased. However, only he one-day-ahead skewness forecass can be economically exploied. This economic significance vanishes once we incorporae ransacion coss. We make hree conribuions o hree respecive srands of lieraure. Firs, we conribue o he lieraure on he predicabiliy of IVSs exraced from index opions. Is findings are mixed depending on he evaluaion meric and daase. Dumas, Fleming, and Whaley (1998) find ha he dynamics of he S&P 500 IVS are highly unsable under an opions hedging seing using weekly daa. On he oher hand, Gonçalves and Guidolin (2006) find ha he S&P 500 IVSs are saisically predicable over daily horizons. These sudies fi complex parameric models o IVSs. Insead, sudying wheher here are predicable paerns in he evoluion of RNMs provides an alernaive and parsimonious way of invesigaing wheher changes in he IVSs can be forecased and hence exends he lieraure on he predicabiliy of IVSs. Furhermore, if RNMs prove o be predicable his will shed ligh on he facor(s) conribuing o he predicabiliy of IVSs. Second, we conribue o he lieraure on he dynamics of higher RNMs exraced from index opions. From a heoreical poin of view, RNMs are expeced o vary over ime due o regime shifs in micro and macro fundamenals (see David and Veronesi (2002), Guidolin and Timmermann (2003)). Alhough here is an exensive empirical lieraure on he dynamics of he second RNM, i.e. he implied volailiy, (see Harvey and Whaley (1992), Konsaninidi, Skiadopoulos, and Tzagkaraki (2008)), surprisingly lile aenion has been paid o he evoluion of risk-neural skewness and kurosis over ime per se. To he bes of our knowledge, he papers by Panigirzoglou and Skiadopoulos (2004) and Lynch and Panigirzoglou (2008) are he closes o ours. Panigirzoglou and 3

4 Skiadopoulos (2004) idenify he facors ha drive he dynamics of he S&P 500 implied probabiliy densiy funcion and assess heir economic significance wihin a value-a-risk seing. Lynch and Panigirzoglou (2008) provide a deailed descripion of he evoluion of RNMs exraced from FTSE 100 and S&P 500 index opions. The auhors examine wheher cross-correlaions beween momens may be useful o forecas he fuure momen values. However, he analysis is performed only wihin sample. Moreover, no oher models are considered for comparison purposes. We ake a more general approach as discussed above. Third, our sudy complemens he vas lieraure on he efficiency of he S&P 500 opion marke (see Broadie, Chernov, and Johannes (2009) for a review of he performance of cerain index opion sraegies). Sana-Clara and Sareo (2009) find ha he profiabiliy of various index opion sraegies largely disappear once marke fricions are aken ino accoun. We add o his lieraure by invesigaing opion rading sraegies based on higher RNMs wih and wihou ransacion coss. The prior lieraure has mosly sudied volailiy rading sraegies (see Konsaninidi, Skiadopoulos, and Tzagkaraki (2008) and references herein). To he bes of our knowledge, Bali and Murray (2011) is he only paper ha proposes a sraegy ha allows aking exposure o he ime variaion of risk-neural skewness. 2 The res of he paper is srucured as follows. Secion II describes he daase. Secion III describes how we exrac he RNMs. The nex secion describes he forecasing models. Secion V presens he in-sample forecasing performance of hese models. Secions VI and VII assess he saisical and economic significance of heir ou-of-sample performance, respecively, over a daily forecas horizon. Secion VIII invesigaes he performance of he models over alernaive forecasing 2 Ai-Sahalia,Wang, and Yared (2001) and Jha and Kalimipalli (2010) also develop higher order RNMs sraegies, ye hese are no based on forecass for he implied momens per se, as we do in his paper. In he former paper, he sraegies are based on deviaions beween implied momens exraced from he cross-secion of marke opion prices and hese exraced from he ime series of he underlying asse in a risk-neural world. In he laer paper, he sraegies are based on deviaions beween implied momens and forecass for he realized momens. Ineresingly, he Chicago Board Opions Exchange (CBOE) inroduced in February 2011 an implied skewness index ha may serve as he underlying o similar ype of skewness sraegies. 4

5 horizons. Secion IX concludes, discusses he implicaions of he resuls, and suggess direcions for fuure research. II. Daa We obain daily S&P 500 European syle index opion daa for January Ocober 2010 from he Ivy DB daabase of OpionMerics. For he purposes of our analysis, we use he S&P 500 implied volailiies provided by Ivy DB for each raded conrac. We selec implied volailiies exraced from opions wih differen srikes and mauriies ranging from five o 270 days. These are calculaed based on he midpoin of bid and ask prices using Meron s (1973) model. In addiion, we obain he closing price of he S&P 500 and he coninuously paid dividend yield from Ivy DB. Following he exising lieraure on calculaing opion-implied disribuions/rnms, we impose several filers on he opion daase prior o exracing RNMs. Firs, we consider only ou-of-he money (OTM) and a-he-money (ATM) opions. Second, we incorporae only opions wih non-zero bid prices and premiums, measured as he midpoin of bes bid and offer, greaer han 3/8 $. Third, we discard opions wih implied volailiies greaer han 100% as well as opions for which Ivy DB does no provide implied volailiies. In addiion, we remove opions wih zero open ineres and zero rading volume. We also discard opions violaing Meron s (1973) arbirage bounds. Finally, we exclude opions ha form verical and buerfly spreads wih negaive prices. U.S. LIBOR raes for mauriies of one o six monhs are obained from Bloomberg as a proxy for he risk-free rae, and he discreely compounded quoes are convered o heir coninuously compounded counerpars. We obain raes for any oher required mauriy by inerpolaing linearly across he adjacen mauriies. In he cases where he desired mauriy is beyond he quoed mauriies, we exrapolae linearly using he closes available LIBOR rae. Finally, we use daa for a se of addiional economic and financial variables o be used as exogenous predicors in our forecasing models. This se comprises he S&P 500 momenum compued as he previous six monh S&P 500 reurn, implied volailiy index VIX downloaded from 5

6 he Chicago Board Opions Exchange (CBOE) websie, rading volume of he shores S&P 500 fuures conrac aken from Daasream, and rading volume of he S&P 500 opions compued from he Ivy DB opion daase. We obain hisorical daa on he 10-year U.S. governmen bond yields and average yields on Moody s BAA and AAA raed corporae bonds from S. Louis Fed websie. III. Exracing risk-neural momens We exrac RNMs from marke opion prices using he model-free mehodology suggesed by Bakshi, Kapadia and Madan (2003, BKM hereafer). A. The BKM mehod Le Q E denoe he expeced value operaor under he risk-neural measure condiional on informaion a ime, r he risk-free rae, C(, ;K ) (P(, ; K ) ) he price of a call (pu) opion wih ime o expiraion τ and srike price K, and R(,τ)=ln[S(+τ)]-ln[S()] he coninuously compounded rae of reurn a ime over a ime period τ. Also le Q r 2 V(, ) E e R(, ) (1) Q r 3 W(, ) E e R(, ) (2) Q r 4 X (, ) E e R(, ) (3) denoe he fair values of hree conracs wih payoff funcions H[S], whereby R(, ) H[S] R(, ) 2 3 (4) 4 R(, ). 6

7 Furhermore, le S( ) e e e S( ) r r r Q r (, ) E ln e 1 V(, ) W(, ) X(, ) be he mean of he log-reurn over period. The risk-neural volailiy (MFIV), skewness (SKEW), and kurosis (KURT) exraced a ime wih horizon τ can be expressed in erms of he fair values of he hree conracs, i.e. MFIV(, ) E R(, ) (, ) V(, )e (, ) (5) Q 2 2 r 2 KURT(, ) SKEW(, ) E ( R(, ) E R(, ) ) Q Q 3 3 Q Q 2 2 E (R(, ) E R(, ) ) e W(, ) 3 (, )e V(, ) 2 (, ) 3 r 2 e V(, ) (, ) 2 r r 3 E ( R(, ) E R(, ) ) Q Q 4 E ( R(, ) E R(, ) ) Q Q 2 2 e X(, ) 4 (, )e W(, ) 6e (, ) V(, ) 3 (, ) r 2 2 e V(, ) (, ) r r r 2 4 (6) (7) The conracs fair values can be deermined by spanning heir payoffs by a porfolio of call and pu opions, as well as he underlying asse and a risk-free bond. I follows ha V(,τ), W(, τ), and X(, τ) can be deermined by a linear combinaion of OTM calls and pus, i.e. K S( ) 2 1 ln S( ) 2 1 ln S( ) K V(, ) C, ;K dk P, ;K dk K K 2 2 (8) S( ) K K S( ) S( ) 6ln 3 ln S( ) 6ln 3 ln S( ) S( ) K K W(, ) C, ;K dk P, ;K dk K K 2 2 (9) S( ) 0 7

8 X(, ) S( ) S( ) K K 12ln 4 ln S( ) S( ) C 2, ;KdK K 2 3 S( ) S( ) 12ln 4 ln K K P,;KdK 2 K (10) B. Empirical implemenaion The implemenaion of equaions (8), (9), and (10) requires a coninuum of OTM call and pu opions across srikes. However, marke opion quoes are available only for a bounded finie range of discree srike prices. This will incur a bias in he calculaion of RNMs (see Dennis and Mayhew (2002), and Jiang and Tian (2005)). In addiion, we need o exrac consan mauriy momens o eliminae he effec of he shrinking ime o mauriy on he dynamics of RNMs as ime goes by. To address boh issues, once we apply he daa filers described in Secion II o any given day, we exrac he expiraions for which a leas wo OTM pus and wo OTM calls are raded. We discard mauriies ha do no saisfy his requiremen. Then, we conver he srike prices of he remaining opions wih a given mauriy ino call delas using Meron s (1973) model. Subsequenly, for any given raded mauriy, in line wih Malz (1997), we inerpolae across he implied volailiies o obain a coninuum of implied volailiies as a funcion of dela. In paricular, we inerpolae on a dela grid wih 1,000 grid poins ranging from 0.01 o 0.99 using a cubic smoohing spline (wih smoohing parameer 0.99). We calculae delas by using he ahe-money (ATM) implied volailiy o ensure ha he ordering of delas is he same as he ordering of srike prices. We discard opions wih delas above 0.99 and below 0.01 as hese correspond o far OTM opions ha are no acively raded. We also ensure ha for each mauriy here are opions wih delas below 0.25 and above 0.75 in order o span a wide range of moneyness regions. If his requiremen is no saisfied, we discard he respecive mauriy from he sample. For delas beyond 8

9 he larges and smalles available dela, we exrapolae horizonally using he respecive boundary implied volailiy. To consruc he consan mauriy momens, we proceed in four seps. Firs, we choose nine dela values (0.1, 0.2,...,0.9), and for each one we inerpolae across he implied volailiies of he various expiraions by using a cubic smoohing spline. Then, from he resuling nine inerpolaed volailiy erm srucures, we selec he respecive implied volailiies for a argeed expiraion. Nex, we obain he consan mauriy implied volailiy curve by fiing a cubic spline hrough hese nine seleced implied volailiies. If he arge expiraion is below he smalles available raded expiraion, a consan mauriy implied volailiy curve is no consruced; exrapolaing in he ime dimension domain yields ime series of implied momens ha exhibi arificially creaed spikes. Finally, we conver he dela grid and he corresponding consan mauriy implied volailiies o he associaed srike and opion prices, respecively, via Meron s (1973) model. Then, we compue he consan mauriy momens [equaions (5), (6), (7)] by evaluaing he inegrals in formulae (8), (9), and (10) using rapezoidal approximaion. We exrac 30, 60, and 90-days consan mauriies o be used subsequenly o verify he robusness of our resuls. C. Risk-neural momens: A preliminary analysis Figure 1 shows he evoluion of he 30, 60, and 90-days consan mauriy S&P 500 RNMs over he sample period. The risk-neural volailiy is annualized. We can see ha he risk-neural skewness is negaive hroughou he enire sample period and for any given mauriy. In addiion, we can see ha in general here is excess kurosis. Boh findings are consisen wih prior lieraure documening ha he index risk-neural probabiliy densiy funcions are negaively skewed and exhibi excess kurosis (see e.g., BKM (2003)). [PLEASE INSERT FIGURE 1 ABOUT HERE] 9

10 For he purposes of our analysis, we divide he sample ino an in-sample par from January January 2000, and an ou-of-sample par spanning he remainder of he daase. Table 1 repors he descripive saisics of he S&P 500 RNMs measured in levels (panel A) and firs differences (panel B) over he in-sample period. The firs order auocorrelaion and values of he augmened Dickey Fuller (ADF) es saisic are repored. We can see ha every RNM is posiively auocorrelaed in he levels, whereas he risk-neural skewness and kurosis are negaively auocorrelaed when measured in firs differences. In addiion, he uni roo es resuls reveal ha all momens (excep for he 30 days implied kurosis) are inegraed of order one. Hence, we will employ he firs differences hroughou our economeric analysis whenever a model requires a saionary ime series. [PLEASE INSERT TABLE 1 ABOUT HERE] D. Risk-neural momens and risk premiums Finally, we examine he relaionship beween he higher RNMs and he variance risk premium (VRP); VRP is defined as he difference beween he risk-neural and saisical expecaions of he fuure reurn variaion (see Bakshi and Madan (2006), Carr and Wu (2009), Bollerslev, Tauchen, and Zhou (2009) for evidence on he VRP exraced from index opions). Exploring his relaionship is ineresing for wo reasons. Firs, Bakshi and Madan (2006) show heoreically ha he VRP is relaed o he physical skewness and kurosis. Based on his resul, Kang, Kim, and Yoon (2010) show heoreically ha he VRP is posiively relaed o he risk-neural skewness and kurosis. Therefore, confirming ha he VRP depends on he risk-neural momens empirically will corroborae he resuls in Bakshi and Madan (2006) and Kang, Kim, and Yoon (2010). Second, Kozhan, Neuberger, and Schneider (2011, Tables 6 and 10) find ha he VRP proxies he skewness risk premium. The profi from buying skew swaps is highly correlaed wih he profi from wriing 10

11 variance swaps. 3 Hence, esing he relaionship beween he VRP and risk-neural skewness and kurosis can shed ligh on he deerminans of he skewness risk premium. To explore he relaionship beween RNMs and VRP, firs we compue he VRP,T a any poin measured over horizons T=30, 60, 90 days for he period January Ocober We do his in order o mach he VRP horizon wih he corresponding horizon of he exraced risk-neural skewness SKEWT, and kurosis T, KURT (see Appendix A for deails on he calculaion of VRP,T ). We find ha on average, he risk-neural expecaion of he fuure reurn variaion is greaer han he saisical one. This is in line wih he findings of he lieraure on VRP exraced from index opions. Subsequenly, we run he following regression for each horizon T 30,60,90 VRP SKEW KURT (11) T, 0 1 T, 2 T, T, Table 2 repors he esimaion resuls for each one of he hree mauriies. We can see ha in all cases he coefficiens of he higher RNMs are posiive and saisically significan. This confirms he relaionship of he higher RNMs o he VRP and hence o he skewness premium. Furhermore, he resuls imply ha an increase in higher RNMs increases he VRP. This is also consisen wih he predicion of he heoreical formula derived by Kang, Kim, and Yoon (2010) on he relaionship beween VRP and higher RNMs. [PLEASE INSERT TABLE 2 ABOUT HERE] IV. The forecasing models We use a number of univariae and mulivariae ime series models o examine wheher RNMs can be prediced. This is because he quesion of predicabiliy is a join hypohesis es of he forecasing model used. The employed models can be grouped ino models ha only uilize informaion from he 3 A skewness swap is a forward conrac ha on he expiry dae pays o is buyer he difference beween he realized (over is life) skewness and he skewness swap rae agreed a he incepion of he conrac. This conrac is analogous o a variance swap where he underlying asse is he realized variance insead of skewness of an asse s reurns. 11

12 RNM ime series per se, and models ha also exploi exogenous informaion from economic and financial variables. We consider he laer class of models because various papers documen he usefulness of exogenous economic and financial variables for predicing volailiy (see Harvey and Whaley (1992), Konsaninidi, Skiadopoulos, and Tzagkaraki (2008)), or condiional asymmery (see Ghysels, Plazzi, and Valkanov (2011)). Nex, we moivae he choice of he paricular economic and financial variables, and hen we describe he models. A. Economic and financial predicor variables We employ he S&P 500 index momenum, lagged rading volume in he S&P 500 fuures and opion marke, pu/call raio, VIX index, measures of he erm spread and defaul spread as well as he lagged risk-free rae as exogenous predicors. Regarding he use of he momenum facor as a predicor, Harvey and Siddique (2000) find ha when pas reurns have been high, skewness is forecased o become more negaive. This phenomenon can be undersood wihin a bubbles heory seing: high pas reurns imply ha he bubble has been inflaing for a long ime and hence here is a larger drop when i burss and prices rever o fundamenals. The burs of he bubble will also affec kurosis because i will yield ouliers and creae a lef fa ail (Blanchard and Wason, 1982). In line wih Harvey and Siddique (2000), we use he previous six monhs S&P 500 reurn o compue he momenum variable. We also include he rading volume as a predicor for RNMs. The heoreical model of Hong and Sein (2003) predics ha skewness becomes more negaive as he rading volume increases. This is because volume proxies for he inensiy of disagreemen among marke paricipans where some of hem face shor sales resricions. The empirical evidence in Hansis, Schlag, and Vilkov (2010) corroboraes he use of rading volume as predicor of subsequen skewness exraced from individual equiy opions. In addiion, Bollen and Whaley (2004), and Gȃrleanu, Pedersen, and Poeshman (2009) propose a demand-based opion pricing approach and find ha measures of he rading volume forecas he shape of he implied volailiy curves, i.e. he RNMs. Noice ha we include he rading 12

13 volume of he underlying asse (proxied by he volume of he shores S&P 500 fuures conrac) as well as he rading volume of he opions as separae predicors. Taylor, Yadav, and Zhang (2009) find ha i is only he volume in he equiy opion markes ha helps explain fuure skewness whereas he volume of he underlying asse does no. Insead, we prefer aking an agnosic view. The use of he pu/call raio as a predicor is jusified because i is regarded as a measure of he marke senimen which affecs RNMs. In paricular, in he case where he marke is pessimisic, he volume of pus is expeced o be greaer han he volume of calls (see Cremers and Weinbaum (2010)). This is because invesors expec he marke o decline and hus he risk-neural probabiliy densiy funcion will appear o be negaively skewed wih a pronounced lef ail. The use of he VIX index is moivaed by he volailiy feedback heory ha implies ha an increase in volailiy increases he index expeced reurns and hence decreases he index price. This resuls in negaive skewness and excess kurosis, oo (see Campbell and Henschel (1992)). The relaionship of he marke senimen and volailiy measures wih risk-neural skewness and kurosis is expeced o be conemporaneous (see Dennis and Mayhew (2002), Taylor, Yadav, and Zhang (2009)), ye i may be se wihin a predicive seing, oo. This is because he pu/call and VIX variables are persisen and hence he duraion of heir effec on higher RNMs is expeced no o have a shor life. We furher employ he erm premium (compued as he difference beween he 10-year and 3- monh U.S. governmen bond yields) and he defaul premium (compued as difference beween he Moody s BAA raed corporae bonds and he AAA raed bonds average yields) as predicors for he RNMs. We employ hese variables because hey serve as leading indicaors for he business cycle. Hence, hey may prove useful in predicing he RNMs ha are also expeced o depend on he business cycle (see Harvey and Whaley (1992)). Finally, we also include he lagged risk-free rae as a predicor for he RNMs. This is because RNMs depend on he mean reurn of he underlying asse, which under he risk-neural measure is he risk-free rae. Given ha he dynamics of ineres raes are commonly modeled as an auoregressive process of order one, he lagged risk-free rae should predic he fuure RNMs. 13

14 B. Univariae ARIMA(X) models We employ univariae auoregressive inegraed moving average models wih (ARIMAX) and wihou exogenous regressors (ARIMA) o sudy he dynamics of RNMs. These models capure he auocorrelaion in RNMs already documened in Secion III.C., as well as auoregressive srucures in he error erms commonly aribued o shor-erm effecs. This choice is also consisen wih papers ha have assumed an auoregressive paern in he dynamics of he respecive physical momens (Harvey and Siddique (1999)). Moreover, he usage of ARIMA models is suppored by he exising lieraure on RNMs. Konsaninidi, Skiadopoulos, and Tzagkaraki (2008) find ha ARIMA models wihou exogenous regressors are useful o forecas implied volailiies; we exend heir analysis o a higher order RNMs seing. In addiion, Chang, Chrisoffersen, and Jacobs (2010) documen ha employing boh auoregressive as well as moving average effecs in modeling RNM series is necessary o remove residual auocorrelaion. The ARIMAX(p,d j,q) model is described by K p d j j(l) MOMENTj, c j j (L) j, k,i Xk, i k1 i1 (12) where d j denoes he order of differencing o produce a saionary and inverible process for he jh momen. In our case, dj 1 for all j as we find all momens o be inegraed of order one. L denoes he lag operaor, p j( L ) 1 j,1l... j,pl and p j(l) 1 j,1l... j,pl denoes he auoregressive and moving average polynomials, respecively, and c j j(1 j,1... j,p ) wih μ j being he mean of ΔMOMENT j,. K denoes he number of exogenous regressors, X k, he observaion of he k h regressor a ime, and ki, he coefficien relaed o he k h regressor a lag i. The case of seing ki, 0 for all k and i yields an ARIMA(p,1,q). In our case, he exogenous regressors are he 14

15 se of variables described in Secion IV.A. To avoid overfiing he daa, we choose p=q=1, and esimae an ARIMA(1,1,1) and an ARIMAX(1,1,1) model. C. Mulivariae models: Vecor Error Correcion Nex, we consider mulivariae ime series models o capure any cross-momens effecs in he dynamics of RNMs. Such effecs have already been documened in he exising lieraure on RNMs exraced from index and equiy opions (see Lynch and Panigirzoglou (2008), Chang, Chrisoffersen, and Jacobs (2010), and Hansis, Schlag, and Vilkov (2010)). From an economeric poin of view, one needs o ake ino accoun he non-saionariy of he RNMs documened in Secion III when formulaing a mulivariae ime series model for he dynamics of RNMs. This is because he RNMs may be coinegraed, ha is a long-run relaionship may exis among hem. In he case where here is coinegraion, a vecor error correcion model (VECM) should be used. To invesigae wheher coinegraion exiss across RNMs, we apply Johansen s (1988) es for coinegraion. We find ha he coinegraion rank is 2, and esimae he following VECM(1) MFIV MFIV MFIV 1 1 SKEW 1 SKEW A1 SKEW 1 KURT 1 KURT KURT 1 1 (13) where A1 is a (3 3) marix of coefficiens conrolling for shor run movemens in RNMs, and Π is a (3 4) marix of coefficiens conrolling he convergence o he long run equilibrium. Π can be decomposed ino a produc of (3 2) marices and, and a (1 2) vecor c such ha Π [ ' c ']; [ 1 2] conains he wo (31) coinegraing vecors 1 and 2, [ 1 2] conains he wo adjusmen vecors, and c conains he consan erms. Given ha he decomposiion of Π is no uniquely defined, we impose he resricions 1 [1 0 13]' and 2 [ ]'. Defining he augmened marix of coinegraing vecors *: c equaion (13) becomes 15

16 MFIV MFIV MFIV 1 1 SKEW1 1 1 KURT 1 KURT KURT 1 SKEW A SKEW *'. 1 (14) Jus as in he ARIMAX case, we also employ a varian of he VECM(1) model by adding exogenous predicors. This yields he following VECM-X(1) model MFIV MFIV MFIV 1 1 SKEW KURT 1 KURT KURT 1 SKEW A SKEW *' X. 1 (15) where X 1 is a (81) vecor ha conains he observaions of he economic and financial variables described in Secion IV.A. measured a ime 1, and is he (3 8) coefficien marix for hese variables. D. Long memory ARFIMA models Finally, we use a univariae ARFIMA(p,d,q) model o invesigae he presence of long-memory effecs. The ARFIMA(p,d j,q) model is defined as d j j( L )(1 L ) ( MOMENT j, j ) j( L ) j, (16) where d j denoes he order of fracional inegraion of he jh momen, (1 L) d j he fracional difference operaor and μ j he expeced value of MOMENTj,. If 0 dj 0.5, hen he process exhibis long memory in he sense ha he sum of he auocorrelaion funcions diverges o infiniy. Again, we se p=q=1 o avoid overfiing he daa. We perform a maximum likelihood esimaion of he ARFIMA(1,d j,1) model in he frequency domain by using he While approximaion of he Gaussian log-likelihood (see Konsaninidi, Skiadopoulos, and Tzagkaraki (2008)). Noe ha in conras o he ARIMA(1,d j,1), he order of (fracional) inegraion for he ARFIMA(1,d j,l) is no 16

17 predeermined. Insead, i is esimaed as par of he esimaion of he model. Hence in general, i will no be he same across momens. Based on he esimaed model, we form one-sep-ahead forecass by aking he infinie auoregressive expansion of he ARFIMA(1,d j,1) process, i.e. E( MOMENT I ) MOMENT ( MOMENT ) (17) j,1 j, j k, j j, k 1 j k1 wih k k i k,j (bi,j jb i1,j )( j ) and i0 b i ( d j i) ( d ) (i 1) j where ( ) denoes he gamma funcion. To implemen equaion (17), we runcae he summaion a 150. V. In-sample evidence We firs esimae all models described in he previous secion using he daily compued RNMs over he in-sample period January 3 rd January 3 rd Table 3 shows he in-sample esimaes for he ARIMA(1,1,1) model across all momens and mauriies. -saisics are repored wihin brackes. We can see ha he adjused R² s are greaer for he risk-neural skewness and kurosis (ranging from 0.11 o 0.19) compared o hese of risk-neural volailiy. Noice ha here is evidence of he exisence of moving average effecs in he RNMs dynamics because he moving average par is saisically significan in mos cases. In addiion, unrepored resuls of he applicaion of he Durbin- Wason es o he residuals of he esimaed ARIMA(1,1,1) models reveal ha he residuals are uncorrelaed for all models and momens. This does no hold uniformly for an AR(1) specificaion. 17

18 [PLEASE INSERT TABLE 3 ABOUT HERE] Table 4 shows he in-sample esimaes for he ARIMAX(1,1,1) model. Adding exogenous regressors o he model slighly improves he goodness of fi compared o he ARIMA specificaion. Regarding he significance of he exogenous regressors, we can see ha he fuures conrac volume and VIX are consisenly significan across mauriies where risk-neural volailiy is forecased. When he higher RNMs are forecased, no predicor is found o be consisenly significan across mauriies. Table 5 shows he esimaion resuls for he VECM(1) wih wo coinegraing vecors for each mauriy. Panel A repors he esimaes of he augmened coinegraing vecors 1 * and * and 2 Panel B repors he esimaes of he adjusmen vecors [ 1 2] and he esimaed coefficiens of marix A 1. The adjused R² of he VECM(1) specificaions for risk-neural skewness and kurosis across he various mauriies are greaer han he ARIMA(X) ones, and range from 0.24 o Hence, he VECM capures he ime variaion in RNMs beer han he ARIMA(X) from a goodness of fi sandpoin. In addiion, mos of he elemens in he adjusmen vecors and he off-diagonal elemens of he coefficien marix A 1 are significan. This implies ha boh he mulivariae srucure of he VECM (i.e. cross-momen effecs), as well as coinegraion predic RNMs in-sample. [PLEASE INSERT TABLE 4 ABOUT HERE] [PLEASE INSERT TABLE 5 ABOUT HERE] [PLEASE INSERT TABLE 6 ABOUT HERE] Nex, we invesigae he incremenal effec of including exogenous regressors o he VECM(1) model. Table 6 shows he esimaion resuls for he VECM-X(1). The inclusion of exogenous regressors increases he adjused R² subsanially for he risk-neural volailiy for all mauriies whereas he effec on he risk-neural skewness and kurosis cases is no uniform across mauriies. 18

19 Similarly o he ARIMAX(1,1,1) model, VIX is he only exogenous predicor ha is consisenly significan across mauriies when i comes o forecasing he risk-neural volailiy. Finally, Table 7 shows he esimaes for he ARFIMA(1,d j,1) model. We can see ha he fracional differencing parameer d j is significanly differen from zero in only a few cases. Moreover, i lies beween 0 and 0.5 for he 60-days risk neural kurosis indicaing ha long-memory effecs may be presen only for his single case. The moving average polynomial parameer θ j is found o be saisically significan in many cases jus as was he case wih he findings for he MA par of he ARIMA(1,1,1) model. [PLEASE INSERT TABLE 7 ABOUT HERE] VI. Ou-of-sample forecasing performance We assess he ou-of-sample forecasing performance of he models described in Secion V over he period January 4 h Ocober 29 h We esimae each model recursively by employing a consan rolling window of daily compued RNMs; he firs esimaion window spans he period January 4 h January 3 rd Subsequenly, a each poin in ime, we form one-day-ahead forecass of he changes of RNMs. We evaluae he accuracy of he obained forecass by hree commonly used measures: (1) he roo mean squared predicion error (RMSE), (2) he mean absolue predicion error (MAE), and (3) he mean correc predicion of he direcion of change (MCP). To evaluae he saisical significance of he obained figures, firs, we examine formally wheher he forecass derived from each one of he considered models ouperform he random walk model ha is seleced as a benchmark. Nex, we idenify he se of bes forecasing models. We implemen all ess separaely for each one of he hree mauriies. A. Relaive performance agains he random walk We use he modified Diebold-Mariano (MDM, Harvey, Leybourne, and Newbold (1997)), and a raio es o assess wheher any model under consideraion ouperforms he random walk model in a 19

20 saisically significan sense under he RMSE/MAE and he MCP merics, respecively. MDM ess he null hypohesis of equal predicive accuracy of a given forecasing model and a benchmark model. Le (e 1,e 2 )denoe he h-sep-ahead forecasing errors of a given model and a benchmark model, respecively, a any poin in ime. These errors are o be evaluaed by a specified loss funcion g(e). Then, he null hypohesis H 0 of equal expeced forecasing performance is H0 E[g(e 1 ) g(e 2 )] 0. (18) Le d g(e 1 ) g(e 2 ) and he sample mean T 1 d T d 1. The variance of d is given by h1 1 V(d ) T 0 2 k k1 (19) where k denoes he k h auocovariance of d and h he forecasing horizon. k is esimaed by n 1 k k k1 ˆ T (d d)(d d) (20) The null hypohesis [equaion (18)] is esed by he MDM es saisic * S1 given by S 1 1/ 2 * T 12hn h(h1) 1 S1 T (21) where 1 1/ 2 S V(d ) d. The criical values of he es are aken from a Suden s disribuion wih (T-1) degrees of freedom. [PLEASE INSERT TABLE 8 ABOUT HERE] Table 8 repors he values for each of he hree error merics for each model, momen, and mauriy over he ou-of-sample period. Regarding he one-day-ahead predicabiliy of he riskneural volailiy, we can see ha he VECM-X(1) ouperforms he random walk under all merics. 20

21 This is in line wih he resuls of Konsaninidi, Skiadopoulos, and Tzagkaraki (2008) who find ha he random walk is ouperformed in forecasing he evoluion of VIX. In he case of risk-neural skewness and kurosis, all models bu he ARFIMA(1,d j,1) ouperform he random walk in almos all cases. B. Choosing he bes models: The model confidence se In conras o he MDM es ha ess only wheher a given model ouperforms a given benchmark, he recenly developed model confidence se (MCS) mehodology by Hansen, Lunde, and Nason (2011) selecs a se of bes forecasing models. Moreover, i acknowledges he informaion conen in he underlying daa in he sense ha uninformaive daa yield a larger se of bes models han informaive daa. To fix ideas, le 0 denoe a se ha conains a finie number of forecasing models where each model is indexed by i 1,...,m0. Each model is associaed wih a series of forecasing errors denoed by ei, and evaluaed by a loss funcion g( ). The performance of model i relaive o he jh model is measured a any poin in ime by he relaive performance variable dij, g(e i, ) g(e j, ) for all i, j 0. The se M * of superior models in M 0 is defined by * i 0 : E( d ij, ) 0 for all j 0. (22) Hence, choosing he bes forecasing model(s) is equivalen o deermining * M. This is done by a sequenial series of significance ess ha remove one inferior model from he se of candidae models in each esing run. Thus, he MCS procedure sars off wih he full se of candidae models 0 and drops models unil i reaches * M. In order o judge saisically wheher M * has been reached or no he following hypohesis ess have o be carried ou in each run H : E( d ) 0 for all i, j vs. H : E( d ) 0 for somei, j (23) 0, ij, A, ij, 21

22 where M M0 is he se of remaining candidae models for * M. If H0, is rejeced in an ieraion a model has o be removed from he se of candidaes and he hypoheses are esed again on he reduced se of models. This procedure is repeaed unil H0, is no rejeced a some predeermined confidence level. The se of models included in a his poin is denoed by as he Model Confidence Se. ˆ * 1 and referred o In order o implemen he MCS procedure, one needs an equivalence es o es H 0,M, as well as an eliminaion rule e ha selecs he model o remove from M if necessary. The equivalence es is based on he range saisic defined as T max (24) i,j ij where ij dij wih var( d ) ij T 1 ij ij, 1 d T d. If he null hypohesis H 0, is rejeced, hen he eliminaion rule given by e arg max sup (25) M im jm ij selecs he wors performing model in M o be removed from he se of candidae models. The asympoic disribuion of he es saisic T under he null hypohesis is non-sandard. Hence, we boosrap he disribuion of he es saisic by implemening he fixed block boosrap algorihm suggesed by Hansen, Lunde, and Nason (2011) o esimae he disribuion of T M under he null hypohesis. We se he fixed block lengh l o a size of 15 observaions and consruc 2,500 boosrap resamples. We find ha resuls from he MCS es are robus o alernaive choices for l. [PLEASE INSERT TABLE 9 ABOUT HERE] We apply he MCS mehodology o all RNMs and mauriies using boh he RMSE and MSE as he error merics. Table 9 repors he resuling MCS p-values for all models, RNMs, and 22

23 mauriies. These p-values have o be inerpreed in he following way: If he p-value for a given model is greaer han some predeermined size of he MCS es, hen his model is included in he -MCS (see Hansen, Lunde, and Nason (2011) for a discussion of he concep of MCS p-values). Two aserisks denoe ha he model is conained in he 5%-MCS under he respecive meric. We can see ha even hough he composiion of he MCS varies across momens and mauriies, he random walk is excluded for all invesigaed mauriies and boh error merics. This finding implies ha RNMs are predicable and is in line wih he findings from he MDM es. Ineresingly, in he case where volailiy is forecased, he MCS is singleon for all mauriies conaining only he VECM- X(1). Our resuls confirm he findings of Gonçalves and Guidolin (2006) who documen ha he IVS exraced from he S&P 500 index opions is predicable over daily horizons. Moreover, our resuls show ha he documened predicabiliy of he IVS can be aribued o no only he implied volailiy bu also o he higher RNMs. VII. Economic significance A. Trading sraegies: Design and implemenaion The resuls on he predicabiliy of he higher RNMs sugges ha heir dynamics can be forecased under a saisical seing. To explore he economic significance of he documened predicabiliy in RNMs, we develop opion rading sraegies (ermed risk-neural or implied skewness and kurosis sraegies) based on he formed ou-of-sample risk-neural skewness and kurosis forecass. We design he proposed sraegies being based on he relaion of he wo higher order RNMs o he shape of he implied volailiy curve measured as a funcion of moneyness on any given day and for any given mauriy (Zhang and Xiang (2008)). In paricular, he slope of he implied volailiy curve is relaed o he risk-neural skewness. The more negaive he risk-neural skewness, he seeper he implied volailiy curve is (see BKM (2003)). Therefore, if he risk-neural skewness is forecased 23

24 o decrease his is equivalen o forecasing ha he implied volailiy skew will seepen and vice versa. The sraegy will amoun o buying OTM pus and selling OTM calls. On he oher hand, he risk-neural (excess) kurosis is relaed o he curvaure of he implied volailiy curve. So for insance, if risk-neural kurosis is forecased o increase, hen his is equivalen o forecasing an increase in he curvaure of he implied volailiy curve. In his case a sraegy would be o sell nearo-he-money opions and buy a-he-money and away from he money opions (see Ai-Sahalia, Wang, and Yared (2001) Figures 9 (a) and (b) for an illusraion of he skewness and kurosis rades in he space of risk-neural probabiliy densiy funcions). Le skew and kur denoe he risk-neural skewness and kurosis, respecively, implied by opion prices a ime and skew, 1 and kur, 1 denoe he one-day-ahead forecas for he risk-neural skewness and kurosis, respecively, ha will prevail a +1 as formed a ime. A any poin in ime, we form he implied skewness rading sraegy based on he following rules: (1) if skew skew, 1 sell all available OTM calls and buy all available OTM pus, (2) if skew skew, 1 buy all available OTM calls and sell OTM pus, (3) if skew skew, 1 do nohing. OTM calls (pus) are defined as having srikes ha are a leas 5% higher (lower) han he underlying price. Similarly, we develop he implied kurosis rading sraegy as follows: (1) if kur kur, 1 buy all available near o-he-money calls and pus and sell all available ATM and deep OTM calls and pus. (2) if kur kur, 1 sell all available near o-he-money calls/pus and buy all available ATM and deep OTM calls and pus, (3) if kur, 1 kur do nohing. Deep OTM calls (pus) are defined as having srikes ha are more han 10% higher (lower) han he underlying price and near OTM calls (pus) have srikes ha are more han 5% higher (lower) bu 10% lower (higher) han he underlying price. ATM calls (pus) have srikes ha are higher (lower) han he underlying price bu less han 5% away from i. We conduc he skewness and kurosis rading sraegies by employing he 60 and 90-days mauriy acual and forecased implied momens; he 30-days mauriy acual and forecased momens 24

25 are no used because here are less daa for hese momens and hence using hem would resul in low rading aciviy. Also, we consider only OTM opions because hese have greaer liquidiy han ITM opions. To immunize he sraegies reurns agains changes in he underlying asse price and is volailiy, we perform boh dela and vega hedging (for a similar approach, see Bali and Murray (2011)). We vega-hedge he opion porfolios using eiher he closes o-he-money shor or long opion in he porfolio as he hedging insrumen depending on wheher a shor or long vega posiion is required o se up he hedge. Subsequenly, we dela-hedge he vega-hedged porfolio by using he S&P 500 as he hedging insrumen in line wih Bakshi, Cao, and Chen (1997). Finally, he reurn on he posiion is compued as he change in he value of he porfolio (opion posiion and underlying posiion) from o +1 divided by he value a. A suble remark is in order a his poin. The exraced RNMs (as well as he forecass) refer o non-raded synheic consan mauriy opions. However, for he purposes of implemening he suggesed sraegies, curren and forecased RNMs ha mach he mauriies of he raded opions should be employed. To address his mismach, we selec he raded expiraion beween 60 and 90 days ha has he greaes number of opion quoes. Nex, a any given ime, we inerpolae linearly across he 60 and 90 days exraced synheic RNMs o obain he (curren) RNMs for he seleced raded expiraion. We also inerpolae across he forecased synheic RNMs o obain he forecased RNMs for he seleced raded expiraion. Then, we compare hese inerpolaed forecased RNMs o he inerpolaed curren RNMs and se up he rades. Noice ha rading does no ake place if here is no available mauriy beween 60 and 90 days. Similarly, if here is no an exraced RNM a or is forecased value for +1, hen again, rading does no ake place. B. Performance evaluaion We evaluae he performance of he rading sraegies on he basis of he Sharpe Raio (SR) and Leland s alpha ( A p, 1999). Ap accouns for he presence of (unrepored) deviaions from normaliy of he disribuion of our sraegies reurns. I is defined as 25

26 Ap E[R Sraegy ] B p(e[r Marke ] R riskfree ) Rriskfree (26) where R sraegy, R Marke, R riskfree, denoe he reurn of he sraegy, marke porfolio, and he risk-free rae, respecively, B p Cov( R Sraegy, (1R Marke ) ) Cov( R (1R ) ) Marke, Marke measures sysemaic risk and ln( E[1R Marke ]) ln( 1R riskfree ) Var[ln( 1 R )] Marke measures he relaive risk aversion. We approximae he reurn on he marke porfolio by he S&P 500 reurn, se he risk-free rae equal o he U.S. LIBOR rae, and esimae A p in a wo-sep procedure. Firs, we compue and B p a each ime sep. Second, we esimae he following regression o deermine he value of A p : RSraegy, B p, (RMarke, R riskfree, ) Rriskfree, Ap (27) If A 0, hen he rading sraegy yields an expeced reurn in excess of is equilibrium risk adjused p level. Table 10 repors he annualized SR and Ap for he risk-neural skewness and kurosis rading sraegies over he ou-of-sample period January Ocober Panels A o E repor resuls for he sraegies implemened based on he daily forecass obained by he respecive models described in Secion IV. The SR of he S&P500 buy and hold sraegy is also repored. To assess wheher he performance measures are saisically differen from zero, we provide boosrapped 95% confidence inervals for he SR and A p o accoun for he non-normaliy in he sraegy reurns disribuion. We calculae hese by employing he Poliis and Romano (1994) saionary boosrap mehod. We se he average block size o en given he (unrepored) low auocorrelaion of he sraegy reurns. We also es alernaive average block sizes and resuls prove o be robus. [PLEASE INSERT TABLE 10 ABOUT HERE] 26

27 We can see ha in he case of rading risk-neural skewness, all models generae economically significan reurns because he SRs and Ap are posiive and significanly differen from zero. This exends he finding repored in Secion VI where hese models ouperform he random walk under he saisical seing. This is because a rading sraegy based on random walk forecass is equivalen o no rading given he rading rules described in Secion VII.A. Moreover, all models ouperform he S&P 500 buy and hold sraegy which yields an SR of Ineresingly, he SRs and Ap are quie similar across models. This may be aribued o he fac ha he values for he MCP meric in Table 8 are also quie similar across models. The MCP meric is he relevan meric in he conex of he skewness and kurosis sraegies as hese are based on he forecased direcion of change in he respecive momens. Wih respec o rading risk-neural kurosis, we can see ha none of he considered models generaes economically significan reurns as he performance measures are no significanly posiive. The evidence ha rading risk-neural skewness yields economically significan profis quesions he efficiency of he S&P 500 opions marke. To fully explore his, we analyze he profiabiliy of he rading sraegies proposed in Secion VII.A by incorporaing ransacion coss. In paricular, we implemen he rading sraegies once more using he quoed bid and ask opion prices provided by Ivy DB. Table 11 repors he performance merics for he wo rading sraegies once ransacion coss are incorporaed. The SR and Ap are negaive for boh he implied skewness and kurosis rading sraegies regardless of he model used o generae he RNM forecass. Hence, he hypohesis of he efficiency of he S&P 500 index opions marke canno be rejeced. This exends he resuls in Gonçalves and Guidolin (2006) who find ha he economic significance of implied volailiy rading sraegies in he S&P 500 opions marke over a one-day horizon vanishes as soon as ransacion coss are incorporaed. [PLEASE INSERT TABLE 11 ABOUT HERE] 27

28 VIII. Furher robusness checks: Alernaive forecasing horizons In his secion, we invesigae wheher he evidence on he predicabiliy of RNMs prevails over longer forecasing horizons. In paricular, we consider weekly and monhly horizons. In order o do so, we form k -sep-ahead forecass by esimaing he forecasing models described in Secion IV wih daily daa, and ieraively forming k-sep-ahead forecass from hese. A ime he forecasing model is esimaed using observaions from -l up o where l denoes he lengh of he esimaion window. Then, one-sep-ahead forecas for he RNMs o prevail a 1 are formed. This forecased value is hen plugged ino he esimaed forecas equaion o obain he 2 forecas. This procedure is repeaed unil he forecas for +k has been obained. A remark is in order a his poin. Despie is simpliciy, he ieraed forecass approach poses one challenge wih regard o he models ha use exogenous predicors. A ime, he informaion only up o ime is known regarding he value of he exogenous predicor. However, in order o form he inermediae RNM forecass up o ime +k, he inermediae values for he exogenous predicors are also required. Hence, hese need o be forecased as well. Given ha any forecasing specificaion for he economic/financial predicors would be ad hoc, we prefer no o include he ARIMAX and VECM-X models in he longer forecas horizon invesigaion. Applicaion of he MDM es for he case of weekly ( k 5 ) and monhly ( k 21) forecass yields he following unrepored resuls. 4 Regarding he weekly forecas horizons, we find ha riskneural skewness and kurosis can be forecased across all mauriies; here exis models ha significanly ouperform he random walk jus as was he case wih he daily horizon forecass. This does no hold for he risk-neural volailiy for which evidence of predicabiliy is raher weak because he random walk is rarely ouperformed. Regarding he monhly forecas horizons, he evidence of predicabiliy is mixed. In paricular, we find ha all 30-days RNMs are unpredicable. On he oher hand, here is evidence of predicabiliy in risk-neural skewness and kurosis for he 60 and 90-days 4 Tables A.1-A.4, found on he journal s websie, conain resuls for he analysis over he longer horizons. 28