The Risk Premia Embedded in Index Options

Transcription

1 The Risk Premia Embedded in Index Opions Torben G. Andersen Nicola Fusari Vikor Todorov March 6, 214 Absrac We sudy he dynamic relaion beween aggregae sock marke risks and risk premia via an exploraion of he ime series of equiy-index opion surfaces. The analysis is based on esimaing a general parameric asse pricing model for he risk-neural equiy marke dynamics using a panel of opions on he S&P 5 index, while remaining fully nonparameric abou he acual evoluion of marke risks. We find ha he risk-neural jump inensiy, which conrols he pricing of lef ail risk, canno be spanned by he marke volailiy (and is componens), so an addiional facor is required o accoun for is dynamics. This ail facor has no incremenal predicive power for fuure equiy reurn volailiy or jumps beyond wha is capured by he curren and pas level of volailiy. In conras, he novel facor is criical in predicing he fuure marke excess reurns over horizons up o one year, and i explains a large fracion of he fuure variance risk premium. We conras our findings wih hose implied by srucural asse pricing models ha seek o raionalize he predicive power of opion daa. Relaive o hose sudies, our findings sugges a wider wedge beween he dynamics of equiy marke risks and he corresponding risk premia wih he laer ypically displaying a far more persisen reacion following marke crises. Keywords: Opion Pricing, Risk Premia, Jumps, Sochasic Volailiy, Reurn Predicabiliy, Risk Aversion, Exreme Evens. JEL classificaion: C51, C52, G12. Andersen graefully acknowledges suppor from CREATES, Cener for Research in Economeric Analysis of Time Series (DNRF78), funded by he Danish Naional Research Foundaion. Todorov s work was parially suppored by NSF Gran SES We are graeful o wo anonymous referees and he edior, Ken Singleon, for many helpful suggesions ha have led o significan improvemens. We also hank Snehal Banerjee, Peer Carr, Anna Cieslak, Bjorn Eraker, Ravi Jaganahan, Bryan Kelly (our discussan a he NBER Meeing), Rober Meron, Sergio Rebelo, Myron Scholes, Ivan Shaliasovich (our discussan in Monreal), Liuren Wu as well as seminar paricipans a Kellogg School of Managemen, Norhwesern Universiy, Duke Universiy, he Monreal 213 Economerics Conference: Time Series and Financial Economerics, he 4 h Annual Meeing of he Danish Economeric Sociey, Sandbjerg, Denmark, he European Universiy Insiue, Florence, 213 Workshop on Measuring and Modeling Financial Risk and High Frequency Daa, he 213 Inernaional Conference in Financial Economerics a Shandong Universiy, Jinan, China, he 4 Years afer he Black-Scholes-Meron Model conference a he Sern School of Business, Ocober 213, he NBER Asse Pricing Meeing a Sanford Universiy, November 213, Georgia Tech, and Boson Universiy for helpful commens. Deparmen of Finance, Kellogg School of Managemen, Norhwesern Universiy, Evanson, IL 628; -andersen@norhwesern.edu. The Johns Hopkins Universiy Carey Business School, Balimore, MD 2122; nicola.fusari@jhu.edu. Deparmen of Finance, Kellogg School of Managemen, Norhwesern Universiy, Evanson, IL 628; v-odorov@norhwesern.edu.

2 1 Inroducion Equiy markes are subjec o pronounced ime-variaion in volailiy as well as abrup shifs, or jumps. Moreover, hese risk feaures are relaed in inricae ways, inducing a complex equiy reurn dynamics. Hence, he markes are incomplee and derivaive securiies, wrien on he equiy index, are non-redundan asses. This parially raionalizes he rapid expansion in he rading of conracs offering disinc exposures o volailiy and jump risks. From an economic perspecive, i suggess ha derivaives daa conain imporan informaion regarding he risk and risk pricing of he underlying asse. Indeed, recen evidence, exploiing parameric models, e.g., Broadie e al. (27), or nonparameric echniques, e.g., Bollerslev and Todorov (211b), finds he pricing of jump risk, implied by opion daa, o accoun for a significan fracion of he equiy risk premium. Sandard no-arbirage and equilibrium-based asse pricing models imply a igh relaionship beween he dynamics of he opions and he underlying asse. This arises from he assumpions concerning he pricing of risk in he no-arbirage seing and he endogenous pricing kernels implied by he equilibrium models. A prominen example is he illusraive double-jump model of Duffie e al. (2) in which he reurn volailiy iself follows an affine jump diffusion. In his conex, he enire opion surface is governed by he evoluion of marke volailiy, i.e., he dynamics of all opions is driven by a single laen Markov (volailiy) process. Recen empirical evidence reveals, however, ha he dynamics of he opion surface is far more complex. For example, he erm srucure of he volailiy index, VIX, shifs over ime in a manner ha is incompaible wih he surface being driven by a single facor. Likewise, Baes (2) documens ha a wo-facor sochasic volailiy model for he risk-neural marke dynamics provides a significan improvemen over a one-facor version. Moreover, Bollerslev and Todorov (211b) find ha even he shor-erm opion dynamics canno be capured adequaely by a single facor as he risk-neural ails display independen variaion relaive o marke volailiy, hus driving a wedge beween he dynamics of he opion surface and he underlying asse prices. The objecive of he curren paper is o characerize he risk premia, implied by he large panel of S&P 5 index opions, and is relaion wih he aggregae marke risks in he economy. As discussed in Andersen e al. (213), he opion panel conains rich informaion boh for he evoluion of volailiy and jump risks and heir pricing. Consequenly, we le he opion daa speak for hemselves in deermining he risk premium dynamics and discriminaing among alernaive hypoheses regarding he source of variaion in risk as well as risk pricing. The sandard no-arbirage approach sars by esimaing a parameric model characerizing he evoluion of he underlying asse price. Risk premia are hen inroduced hrough a pricing 1

3 kernel which implies ha risk compensaion is obained hrough parameer shifs. This ensures, convenienly, ha he associaed risk-neural dynamics remains wihin he same parameric class enerained for he saisical measure, see, e.g., Singleon (26), Chaper 15. However, his approach ends o ie he equiy marke and opion surface dynamics closely ogeher. In paricular, he equiy risk premia are ypically linear in (componens of) volailiy. In conras, we find he opions o display risk price variaion ha is largely unrelaed o, and effecively unidenifiable from, he underlying asse prices alone. This moivaes our reverse approach of saring wih a parameric model for he risk-neural dynamics and esimaing i exclusively from opion daa along wih no-arbirage resricions based on model-free volailiy measures consruced from he underlying asse daa. In his manner, we avoid leing a (possibly misspecified) parameric srucure for he P-dynamics impac he idenificaion of opion risk premia. We hen explore he risk premia dynamics by combining he exraced sae vecor wih high-frequency daa on he equiy index. We documen ha even a very general wo-facor sochasic volailiy model wih jumps boh in price and volailiy as well as a ime-varying jump inensiy produces sysemaic biases in he fi o he opion surface. The problems are paricularly acue following periods like he Asian crisis in 1997 or he grea recession originaing in he Fall of 28. Afer hese evens, marke volailiy revers back owards is pre-crisis level fairly quickly, while ou-of-he-money pu opions remain expensive, inducing a seepening of he (Black-Scholes) implied volailiy curve. This ype of variaion in he opion surface is difficul o accommodae for sandard (no-arbirage or equilibrium) models as hey imply ha he priced jump ail risk, which in urn deermines he ou-of-he-money pu prices, is governed solely by (componens of) marke volailiy. This feaure implies, in paricular, ha hese models end o generae risk premia ha are oo low in he afermah of crises. We follow Andersen e al. (213) by inroducing a hird facor driving he risk-neural jump inensiy. This novel ail facor is no par of he volailiy dynamics alhough i may be correlaed wih he level of volailiy. We model i as purely jump driven, wih one componen joinly governed by he volailiy jumps while anoher is independen of he volailiy process. This feaure allows he jump inensiy o escalae hrough so-called self-exciaion of he jumps in periods of crises when price and volailiy jumps are prevalen, hus magnifying he response of he jump inensiy o major (negaive) marke shocks. The exended model remains wihin he popular class of affine jump-diffusion models of Duffie e al. (2) and exemplifies he flexibiliy of such models for generaing inricae, ye analyically racable, dynamic ineracions beween volailiy and jump risks. This ype of exension has no been explored in prior empirical opion pricing sudies. 2

4 Relaive o he empirical analysis in Andersen e al. (213), we esimae he model by minimizing, no he squared, bu he relaive squared opion pricing error across he full sample. This reduces he weigh assigned o highly urbulen periods where he bid-ask spreads and pricing errors increase sharply. We find ha he ail facor improves he characerizaion of he opion surface dynamics very significanly. In paricular, he new model no longer undervalues shor-mauriy ou-of-he-money pus in he afermahs of crises. Hence, our exended risk-neural model provides a more suiable basis for sudying he dynamic properies of marke risk premia. In urn, he presence of his independenly evolving ail facor implies ha par of he risk premium dynamics canno be capured by he sae variables driving he underlying asse price dynamics. This implies ha his jump risk facor may have predicive power for fuure risk premia over and above wha is implied by he volailiy facors. This is indeed wha we find. The novel ail facor is significan in forecasing fuure excess marke reurns for horizons up o one year while he volailiy facors are insignifican. Similarly, he new ail facor is imporan for predicing he fuure variance risk premium in conjuncion wih one of he volailiy componens. Taken ogeher, our findings raionalize why he variance risk premium provides superior forecass for fuure reurns relaive o volailiy iself, as documened in Bollerslev e al. (29). The key is he exisence of he new facor driving he lef jump ails of he risk-neural disribuion. Imporanly, while he new jump facor has predicive power for risk premia, i conains no incremenal informaion regarding he fuure evoluion of volailiy and jump risks for he underlying asse relaive o he radiional volailiy facors. Hence, our findings indicae ha opion markes embody criical informaion abou he marke risk premia and is dynamics which is essenially unidenifiable from sock marke daa alone. Moreover, he opion surface dynamics conains informaion ha can improve he modeling and forecasing of fuure volailiy and jump risks, bu such applicaions necessiae an iniial unangling of he componens in he risk premia ha evolve independenly from he volailiy process. Overall, our empirical resuls sugges ha here is a wedge beween he sochasic evoluion of risks in he economy and heir pricing, wih he laer ypically having a far more persisen response o (negaive) ail evens han he former. Our finding of a subsanial wedge beween he dynamics of he opion and sock markes presens a challenge for radiional srucural asse pricing models. Specifically, he sandard exponenially-affine equilibrium models wih a represenaive agen equipped wih Epsein-Zin preferences imply ha he raio of he risk-neural and saisical jump ails is consan. On he conrary, he new facor, exraced from he opion daa, drives he risk-neural jump ail bu has no discernable impac on he saisical jump ail. We conjecure ha his wider gap beween 3

5 fundamenals and asse prices may be accouned for hrough an exension of he preferences via some form for ime-varying risk aversion and/or ambiguiy aversion owards exreme downside risk. The res of he paper is organized as follows. Secion 2 describes our daa. Secion 3 presens our exended hree-facor model which, combined wih flexibiliy in he modeling of he jump disribuions, encompass mos exising models in he empirical opion pricing lieraure. Secion 4 inroduces he esimaion mehodology and discusses he parameer esimaes and he formal diagnosic ess for he fi o he opion surface. Secion 5 explores alernaive diagnosics for model fi and differen robusness checks. This analysis brings ou some of he mechanisms behind he improved fi of our hree facor model. Secion 6 is dedicaed o an ou-of-sample analysis, documening he robusness of he esimaion resuls and inference. In Secion 7 we exploi he esimaion resuls o sudy he risk premium dynamics and is implicaion for reurn and variance predicabiliy. Our findings are conrased o corresponding predicabiliy resuls implied by popular srucural equilibrium models. Secion 8 concludes. In a Supplemenary Appendix we repor esimaion resuls on subsamples as well as for various alernaive specificaions for he risk-neural dynamics of he underlying index. The Supplemenary Appendix also conains addiional diagnosic ess relaed o he parameric fi for he opion surface. 2 Daa and Preliminary Analysis We use European syle S&P 5 equiy-index (SPX) opions raded a he CBOE. We exploi he closing bid and ask prices repored by OpionMerics, applying sandard filers and discarding all in-he-money opions, opions wih ime-o-mauriy of less han 7 days, as well as opions wih zero bid prices. For all remaining opions, we compue he mid bid-ask Black-Scholes implied volailiy. The daa spans January 1, 1996 April 23, 213. I is furher divided ino an in-sample period covering January 1, 1996 July 21, 21, and an ou-of-sample period consising of July 22, 21 April 23, Following earlier empirical work, e.g., Baes (2) and Chrisoffersen e al. (29), we sample every Wednesday. 2 The in-sample period includes 76 rading days, and he esimaion is based on an average of 234 bid-ask quoes per day The ou-of-sample period conains 142 rading days and feaures subsanially more quoes, and we end up exploiing an average of 78 opion conracs per day from his sample. The nonparameric esimae of volailiy used for penalizing he objecive funcion below is consruced from one-minue daa on he S&P 5 fuures covering he ime span of he opions. The same daa are used o consruc measures of volailiy 1 This is a rue ou-of-sample period as he enire analysis was compleed for he in-sample period before we obained he addiional daa covering he more recen years. 2 Due o exreme violaions of no-arbirage-condiions, we replaced Ocober 8, 28, wih Ocober 6, 28. 4

6 and jump risks for he predicive regressions in Secion 7. Finally, we employ he reurns on he SPY ETF raded on he NYSE, which racks he S&P 5 index porfolio, and he 3-monh T-bill rae o proxy for he risk-free rae, when implemening hese regressions. 2.1 The Opion Panel We denoe European-syle ou-of-he-money opion prices for he asse X a ime by O,k,τ. Assuming fricionless rading in he opions marke, he opion prices a ime are given as, O,k,τ = E Q E Q [ e ] +τ r s ds (X +τ K) +, if K > F,+τ, [ e ] +τ r s ds (K X +τ ) +, if K F,+τ, where τ is he enor of he opion, K is he srike price, F,+τ is he fuures price for he underlying asse a ime referring o dae + τ, for τ >, k = ln(k/f,+τ ) is he log-moneyness, and r is he insananeous risk-free ineres rae. Finally, we denoe he annualized Black-Scholes implied volailiy corresponding o he opion price O,k,τ by κ,k,τ. This merely represens an alernaive noaional convenion, as he Black-Scholes implied volailiy is a sricly monoone ransformaion of he raio er,+τ O,k,τ F,+τ, where r,+τ denoes he risk-free rae over he period [, + τ]. The empirical work explicily accouns for measuremen error in he opion prices. We denoe he average of he bid and ask quoes (expressed in Black Scholes implied volailiy unis) by κ,k,τ, and view his as a noisy observaion of underlying value. To he exen he measuremen errors are no srongly correlaed across a large fracion of he surface, we improve he efficiency of he inference by incorporaing he full opion cross-secion in our esimaion and esing procedures, effecively averaging ou idiosyncraic observaion errors. The size of he spread varies over ime and is posiively correlaed wih he volailiy level. In addiion, here are sysemaic differences in he relaive spread across moneyness. For example, he spread is abou 8% of he mid-spread level for deep OTM pus, on average, implying ha a ypical implied volailiy (IV) reading of 4% is associaed wih bid and ask quoes of 38.4% and 41.6%. Similarly, for an IV of 18% for ATM opions, he quoes are generally around 17.6%-18.4%, while a ypical se of quoes for far OTM calls are 18.8%-21.2% for a mid-poin value of 2%. The opions underlying he implied volailiy (IV) surface are highly heerogeneous in erms of moneyness and enor across ime. To faciliae comparison, we creae a uniform se of regions based on he opion characerisics. Specifically, we define he volailiy-adjused moneyness, m, a ime for enor τ, by sandardizing he log-moneyness wih he a-he-money implied volailiy, (2.1) m = k κ,,τ τ. 5

7 Table 3 shows how he observaions in our sample are disribued across he opion surface. The four regions of moneyness represen deep OTM pu opions, OTM pu opions, ATM opions and OTM call opions, while he wo caegories for ime-o-mauriy provide a rough spli ino shor versus long daed opions. No surprisingly, here is paricularly good coverage for ATM opions, which represen over 44% of he in-sample observaions. The quoes for he OTM call opions are somewha limied and amoun o almos 16% of he oal opions quoes, roughly maching he proporion of deep OTM pu opions. Finally, over 24% fall in he OTM pu opion region. 1996:1-21:7 21:7-213:4 τ 6 τ > 6 τ 6 τ > 6 m < m < m m > Table 1: Relaive Number of conracs. We repor he percenage of opion conracs ha, on average, fall wihin he differen combinaions of moneyness and enor for he indicaed sample. In he ou-of-sample period, he daily number of acive quoes is much higher, especially for deep OTM opions. Since we would like o compare model performance across he wo samples, we have runcaed he se of OTM opions included in he recen period o lie wihin he boundaries of -7.1 for he pus and 2.4 for he calls. These cu-offs correspond o he average minimum and maximum moneyness of he opion quoes for our in-sample period. This sandardizaion limis he heerogeneiy across he samples. The relaive proporion of OTM calls and pus is sable, bu we observe a non-rivial shif from ATM o deep OTM pu opions, wih he former now represening abou 32.5% and he laer 27% of he overall observaions. To he exen variaion in he pricing of deep OTM pu opion prices is harder o accommodae han for ATM prices, his composiion effec will all else equal imply a worse ou-of-sample fi han would oherwise be observed. The opion IV surface displays a highly persisen and nonlinear dynamic which is difficul o convey effecively hrough a few summary measures. We provide a couple of alernaive depicions ha highligh differen aspecs of he dynamic. The firs approach emphasizes he evoluion of separae surface characerisics, as represened by he (annualized) IV level, erm srucure, skew, and skew erm srucure. These quaniies are ploed in Figure 1. Our second approach consiss of a sandard principal componen analysis of he IV surface, and we explore hese in deail below. 6

8 In Figure 1, he level capures he average IV for ATM shor-daed opions, he erm srucure reflecs he difference beween he IV of long and shor mauriy ATM opions, he skew measures he IV gap beween shor-daed OTM pu and OTM call opions, and, finally, he skew erm srucure is he difference beween he skew compued from long- and shor-daed opions. The IV level displays occasional erraic spikes o he upside, bu also displays srong persisence, as expeced for a series reflecing he general level of volailiy. Inspecing he remaining hree panels, he degree of commonaliy is sriking. Every major spike in he IV level is visible in he oher characerisics, albei in he opposie direcion for he IV and skew erm srucures. Tha is, rapidly rising volailiy is accompanied by a downward sloping erm srucure, a seepening of he (shor) IV skew, and a sharp increase in he shor versus long skew. Table 2 supplemen Figure 1 wih summary saisics for he IV surface characerisics. The correlaion marix confirms he srong covariabiliy beween he IV level and he remaining feaures. Moving o he individual characerisics, we see ha he IV erm srucure is moderaely posiive, apar from episodic large negaive ouliers. The skew is consisenly srongly posiive, exceeding 1% over he vas majoriy of he sample and averaging 18%. The skew also displays srong persisence and is paricularly elevaed when markes are urbulen. Finally, he skew erm srucure is negaive for almos he enire sample, so he smirk flaens subsanially wih opion enor. This laer feaure accouns for a grea deal of variaion in he surface shape, as he skew is exremely negaively correlaed wih he skew erm srucure. Thus, when volailiy soars and he skew seepens, he effec is ypically much less pronounced a he longer mauriies. Hence, his ype of exciemen of he shor lef par of he IV surface mus be associaed wih shocks o volailiy and jump inensiies of low o moderae persisence, as he effec is srongly a longer horizons. This is also consisen wih he covariaion of he IV and skew erm srucures as well as he comparaively mild negaive associaion beween he skew and IV erm srucure. Turning o he principal componen (PC) analysis, Figure 2 depics he in-sample realizaions of he four firs PCs. I is eviden ha he firs PC is closely relaed o he IV level, while he second PC displays commonaliy wih he IV erm srucure. However, he las wo PCs appear largely unrelaed o he characerisics depiced in Figure 1. The firs PC capures abou 95.7% of he oal variaion, while he following PCs accoun for, respecively, 2.3%,.7% and.3%. Clearly, here is a dominan level ype effec, bu his facor also accouns for a grea deal of variaion in he skew, erm srucure and skew erm srucure, leaving, relaively speaking, only minor residual variaion o explain for he remaining PCs. 7

9 .7 Implied Volailiy Level.1 Implied Volailiy Term Srucure Jan98 Jan Jan2 Jan4 Jan6 Jan8 Jan1.25 Jan98 Jan Jan2 Jan4 Jan6 Jan8 Jan1.8 Implied Volailiy Skew.1 Implied Volailiy Skew Term Srucure Jan98 Jan Jan2 Jan4 Jan6 Jan8 Jan1.6 Jan98 Jan Jan2 Jan4 Jan6 Jan8 Jan1 Figure 1: Implied volailiy (IV) surface characerisics. Top lef Panel: IV Level defined as he average IV for shor ATM opions. Top Righ Panel: IV Term Srucure defined as he difference in IV beween long and shor ATM opions. Boom lef Panel: IV skew defined as he difference beween IV of shor OTM pu and OTM call opions. Boom righ Panel: skew erm srucure, defined as he difference beween long and shor skew, where he long skew is defined as he shor skew, bu using long-daed opions. We define shor opions as hose wih less han 21 days o mauriy and long opions as hose wih more han 18 days o mauriy. We define ATM opions wih volailiy-adjused moneyness m wihin [.2,.2]; OTM pu opions wih m < 3 (shor mauriies) and m < 1.5 (long mauriies); OTM call opions wih m > 1. Summary saisics Correlaion Marix Mean Median Sd Level TS Skew Skew TS Level TS Skew Skew TS Table 2: Summary saisics and correlaion marix for surface characerisics. The saisics for he implied volailiy Level, Term Srucure (TS), Skew, and Skew Term Srucure are compued over he in-sample period, January 1996 July 21. 8

10 6 PC1: % 4 PC2: % Jan98 Jan Jan2 Jan4 Jan6 Jan8 Jan1 8 Jan98 Jan Jan2 Jan4 Jan6 Jan8 Jan1 1 PC3:.65684% 8 PC4:.31394% Jan98 Jan Jan2 Jan4 Jan6 Jan8 Jan1 6 Jan98 Jan Jan2 Jan4 Jan6 Jan8 Jan1 Figure 2: Principal Componens of he Implied Volailiy Surface. The panels depic he firs four principal componen exraced from he S&P 5 implied volailiy surface from January 1996 o July 21. On each day we inerpolae he IV surface o generae sandardized opions wih he same volailiy-adjused moneyness, m, and enor across he sample. Specifically, we obain opion prices for m in he se { 4, 3, 2, 1,, 1} and enor equal o hree values, {.1,.3, 1} (in years). This produces a oal of 18 synhesized opion conracs per day. For each panel, we also repor he percenage of he overall variaion explained by he given principal componen. 9

11 To furher explore how he PCs inerac wih he IV surface characerisics, Table 3 repors on he in-sample regression of characerisics on PCs. The able confirms he srong associaion of he firs PC wih he IV level, feauring a -saisic beyond 1. For his PC, we also obain he negaive associaion wih he IV and skew erm srucures and posiive relaion wih he skew, maching he correlaion paerns for he IV level in Table 2. In addiion, we find he second PC o be associaed wih he IV erm srucure, corroboraing he visual impression from Figure 2. The hird PC is weakly associaed wih he skew as well as he skew erm srucure and level. However, he skew is much more direcly associaed wih he firs PC so, effecively, he hird PC capures only residual skew variaion ha is largely orhogonal o he level. Finally, he fourh PC has no idenifiable relaion o any of he characerisics. In paricular, he variaion in he skew erm srucure is effecively accouned for hrough he firs wo or hree PCs. Level TS Skew Skew TS PC1.19 (147.92) -.5 (-34.1).19 (66.82) -.9 (-27.52) PC (-32.92).46 (51.98).1 (.32).45 (2.57) PC (-11.11) -.7 (-4.2).64 (18.34) -.43 (-1.52) PC4.14 (6.13) -.14 (-5.7) -.45 (-8.9) -.2 (-.34) R AC(1) AC(2:1) AC(11:2) Table 3: Relaing Opion Characerisics and Principal Componens. We repor he coefficiens wih -saisics (in parenheses) and he R 2 from he linear regression of differen opions characerisics (level, erm srucure, slope, and slope erm srucure) on he firs four principal componens exraced from he S&P 5 implied volailiy surface from January 1996 o July 21. We also repor he firs sample auocorrelaion coefficien and he average sample auocorrelaion coefficien over wo o en and eleven o weny lags of he regression residuals. These observaions have a number of implicaions. Firs, here is no simple mapping beween PCs and characerisics. The laer are inrinsically inerconneced and covary srongly. Hence, i is impossible o associae a specific facor wih feaures of a given characerisic. Such facors will generally exer a significan join impac on muliple characerisics. Second, he difficuly of separaing he forces driving he individual characerisics is, of course, a general feaure of sysems wih pronounced nonlineariies. We see furher evidence of poenial nonlineariies in he 1

12 large degree of serial correlaion in he residuals of he regression in Table 3. The firs order auocorrelaion coefficien is very large across all he characerisics, and hey die ou exremely slowly, suggesing highly persisen deviaions from he linear approximaion associaed wih he regression analysis. In shor, he IV surface dynamics is likely highly nonlinear in he underlying facors. 3 Third, he persisen residuals can be inerpreed as evidence for missing facors and, indeed, we found sandard ess for he number of facors, developed in, e.g., Bai and Ng (22), o indicae 7 o 8 facors. Given he nonlinear associaion beween opion IV and facors, his likely reflecs he failure of he linear approximaion raher han an indicaion of he rue number of underlying (linear) facors. Fourh, given he nonlinear relaion beween he he IV surface and underlying facors, i is problemaic o associae he firs few (linear) PCs wih acual facors. 4 In summary, accouning for a given model s abiliy o rack he IV surface dynamics provides an inuiive and useful way o highligh he implicaions of model misspecificaion. Indeed, we laer illusrae he qualiy of our own model fi by comparing he model-implied evoluion of he characerisics o he IV surface characerisics generaed by he daa. On he oher hand, here is no direc associaion beween he inabiliy o fi he dynamics of a specific se of characerisics and he lack of a given facor, because he facors impac on he opion IVs are highly nonlinear and creae correlaed dynamic ineracions across he characerisics. Laer in he paper, we exploi new diagnosic ools for he model fi o broader regions of he IV surface, which provide a more formal basis for guiding model specificaion and selecion. 3 Parameric Modeling of he Opion Panel We adop a general-o-specific approach o he parameric modeling of he risk-neural equiy-index reurns, wih he objecive of capuring he salien feaures of he IV surface dynamics succincly. The model subsequenly serves as he basis for our analysis of he equiy and variance risk premiums. The main iniial decision concerns he number of laen sae variables o include. The vas majoriy of empirical opion pricing sudies employs a single sochasic volailiy facor, bu he lieraure on esimaing he reurn dynamics under he physical measure as well as a few opion pricing sudies, e.g., Baes (2), Chrisoffersen e al. (29), Chrisoffersen e al. (212), and Andersen e al. 3 For a simple illusraion, we calibraed a one-facor Heson model and derived he sensiiviy of he IV surface o he volailiy facor. For a given locaion in he surface, i.e., a given mildly OTM opion, he sensiiviy (derivaive of he opion IV wih respec o volailiy) will ypically vary dramaically wih he level of volailiy. Thus, srongly correlaed approximaion errors will arise from linearizing he dependence of he surface characerisics o facors, as he rue sensiiviies flucuae srongly over ime. This illusraion is provided in our supplemenary web appendix. 4 In his respec, he dynamics of he IV surface deviaes sharply from ha of he he erm srucure of ineres raes. For he laer, he firs hree PCs exraced from he variaion in he yields across he available mauriies provide excellen linear spanning for he full yield curve. 11

13 (213), poin o a minimum of wo facors. This is also consisen wih our descripive analysis of he IV surface in Secion 2 which found he characerisics o load srongly on he firs wo principal componens and quie significanly on he hird PC as well. Hence, we follow Andersen e al. (213) in proposing a general hree-facor model which, apar from he specificaion of he jump disribuion, embeds all exising coninuous-ime models in he lieraure as special cases. We documen below ha exponenially disribued price jumps provide a superior fi relaive o he more commonly adoped Gaussian specificaion, jusifying he alernaive represenaion in our benchmark model. Afer eliminaing hose feaures, ha are joinly economically and saisically insignifican, we arrive a a racable and relaively parsimonious hree-facor model ha provides a saisfacory fi o he opion surface, boh in- and ou-of-sample. Our hree-facor model for he risk-neural equiy index dynamics is given by he following exension of he model proposed in Andersen e al. (213), dx = (r δ ) d + V 1, dw Q 1, X + V 2, dw Q 2, + η U dw Q 3, + (e x 1) µ Q (d, dx, dy), R 2 dv 1, = κ 1 (v 1 V 1, ) d + σ 1 V1, db Q 1, + µ 1 x 2 1 {x<} µ(d, dx, dy), dv 2, = κ 2 (v 2 V 2, ) d + σ 2 V2, db Q 2,, du = κ u U d + µ u R 2 [ (1 ρu ) x 2 1 {x<} + ρ u y 2] µ(d, dx, dy), where (W Q 1,, W Q 2,, W Q 3,, BQ 1,, BQ 2, (W ) is a five-dimensional Brownian moion wih corr Q 1,, BQ 1, = ρ 1 ( ) and corr W Q 2,, BQ 2, = ρ 2, while he remaining Brownian moions are muually independen. In addiion, µ is an ineger-valued couning variable (random measure on R + R 2 ), represening he jumps in he price, X, as well as he sae vecor, (V 1,, V 2,, U ). The corresponding (insananeous) jump inensiies, under he risk-neural measure, is d ν Q (dx, dy). This process is also known as he (jump) compensaor, and he difference beween he jump realizaions and he compensaor, µ Q (d, dx, dy) = µ(d, dx, dy) dν Q (dx, dy), consiues he associaed maringale (jump) measure. R 2 The jump specificaion involves wo separae componens, x and y. The former capures cojumps ha occur simulaneously in he price, he firs volailiy facor, V 1,, and, poenially, in he U facor (if ρ u < 1), while he y jumps represen independen shocks o he U facor. The laer can also generae a jump in reurn volailiy (if η > ), bu he main effec is hrough he jump inensiies. The compensaor characerizes he condiional jump disribuion. I is given by, ) (3.1) ν Q (dx, dy) dxdy = ( c () 1 {x<} λ e λ x + c + () 1 {x>} λ + e λ+x) 1 {y=} + c () 1 {x=, y<} λ e λ y. 12

14 The firs erm on he righ hand side, referring o he x, y = case, reflecs co-jumps in price and volailiy, while he second erm, x =, y <, capures independen shocks o he U facor. Hence, he individual (sricly posiive) jumps in U are eiher independen from V 1 or proporional o he (simulaneous) jump in V 1. The price jumps are exponenially disribued, wih separae ail decay parameers, λ and λ +, respecively, for negaive and posiive jumps. Moreover, for parsimony, he independen shocks o he U facor is disribued idenically o he negaive price jumps. Finally, he ime-varying jump inensiies are governed by he c () and c + () coefficiens. These coefficiens evolve as affine funcions of he sae vecor, c () = c + c 1 V 1, + c 2 V 2, + U, c + () = c + + c+ 1 V 1, + c + 2 V 2, + c + u U. This represenaion involves a large se of parameers ha can be hard o idenify separaely. A he esimaion sage, we eliminae hose ha are insignifican and have no discernable impac on model fi. However, generaliy along his dimension is imporan, as he jump specificaion urns ou o be criical for a suiable represenaion of he IV surface dynamics. Our hree-facor model possesses a number of disincive feaures. The facors (V 1, V 2, U) drive boh he diffusive volailiy and he jump inensiies. V 1 and V 2 are always presen in he diffusive volailiy, as in radiional muli-facor volailiy models, while U conribues o diffusive volailiy only if η >. In fac, he consrained model arising for η = (or very small) is of separae ineres. I implies ha he facor U affecs only he jump inensiies, wih no (lile) impac on diffusive volailiy. Furhermore, in an exension o exising opion pricing models, we allow posiive and negaive jump inensiies o have differen loadings on he laen facors. In paricular, some facors affec only posiive or only negaive jump inensiies. Such flexibiliy in modeling he jump inensiies is imporan given he nonparameric evidence in Bollerslev and Todorov (211b). The jump modeling also involves several disincive feaures. Firs, he price jumps are exponenially disribued. This is unlike mos earlier opion pricing sudies which rely on Gaussian price jumps, following Meron (1976). Nonparameric evidence in Bollerslev and Todorov (213) suggess he exponenial price jumps capure he behavior of shor-mauriy OTM pus much beer, and we confirm his in our empirical analysis. Nex, he jumps in he facor V 1 is linked deerminisically o he price jumps, wih squared price jumps impacing he volailiy dynamics in a manner reminiscen of GARCH models. For parsimony and ease of idenificaion, we allow only he negaive price jumps o impac he volailiy dynamics. 5 Finally, U is driven in par by he (same) squared negaive price jumps and in par by independen jumps, wih he parameer ρ u conrolling 5 Suppor for his ype of price-volailiy jump dependence, albei under P, may be found in Todorov (211). 13

15 he conribuion of each componen in he dynamics of U. As such, he model accommodaes boh perfec dependence (ρ u = ) and full independence (ρ u = 1) beween he jump risks of V 1 and U. Moreover, hese sae variables, governing imporan feaures of he opion surface dynamics, are relaed hrough he ime-variaion in he jump inensiy. Our specificaion allows for cross self-exciaion in which jumps in V 1 enhance he probabiliy of fuure jumps in U, and vice versa. 6 Differences in he jump disribuions aside, our model ness mos exising models as special cases. In paricular, in he one-facor seing, wih V 2 and U absen, we recover he double jump volailiy model of Duffie e al. (2), esimaed using opion daa by Broadie e al. (27). In he wo-facor seing, wih U absen and excluding volailiy jumps (in V 1 ), we obain he Baes (2) jump-diffusion, and furher ruling ou price jumps leads o he wo-facor diffusive model of Chrisoffersen e al. (29). Finally, our jump dynamics shares some qualiaive feaures wih he discree-ime model in Chrisoffersen e al. (212). Consequenly, we generalize exising wofacor coninuous-ime models by incorporaing heir main feaures joinly, while furher allowing for volailiy jumps, an ingredien sressed by Broadie e al. (27). Finally, we inroduce he new jump-driven facor, U, which accommodaes addiional nonlineariies in he opion surface dynamics, while sill reaining he racabiliy afforded by he exponenially-affine seing. The main deparure from prior work sems from he inclusion of he new U facor. Given he raher unconvenional represenaion, we briefly discuss how his facor enhances he feaures of he risk-neural dynamics. I is bes seen by focusing on a resriced version of he model where we eliminae U from he diffusive volailiy, i.e., se η =, and furher le ρ u =, so ha here is no independen jump componen driving U. In his scenario, here is no disinc source of risk impacing U. I is driven solely by he squared negaive price jumps. Noneheless, since U affecs he jump inensiies separaely from V 1, i may sill influence he opion surface dynamics in criical ways. Tha is, o convey he curren sae of he sysem, U mus be included among he componens of he sae vecor. 7 In oher words, even if he source of risk in U is spanned by he jumps in X and V 1, U is sill necessary for characerizing he disribuion of fuure log reurns or predicing he fuure evoluion of he facors, even afer conrolling for he curren values of V 1 and V 2. Of 6 We sress ha he model (3.1) sill belongs o he affine family covered by Duffie e al. (23) and, as shown in Andersen e al. (213), he following parameer consrains ensure covariance saionariy of he laen facors, κ 1 > 2c 1 µ1 2c 3, and κ λ 2 3 > κ1µu κ 1λ 2, and κ 2 <, and σ 2 2c 1 µ1 i 2κ iv i, i = 1, 2. 7 This is a reflecion of he fac ha here, in general, is no direc associaion beween he dimensionaliy of he sources of risk and he dimension of he sae vecor required o characerize he condiional dynamics of he sysem. This phenomenon arises in coninuous-ime ARMA model, see, e.g., Brockwell (21). I is also a well-known feaure of he so-called quadraic erm srucure models, see, e.g. Ahn e al. (22) and Leippold and Wu (22). 14

16 course, he role of U only expands, if i is subjec o independen shocks as well. In he empirical secion below, we deail how he U facor impacs he IV surface characerisics over ime. The model in equaion (3.1) perains o he risk-neural dynamics of X. However, due o he equivalence of Q and P, he assumed dynamics have implicaions for he dynamics of X under P as well. In general, hese implicaions are limied o hose feaures of model (3.1) which hold almos surely. They consis of he following. Firs, he spo diffusive variance is invarian o he change of measure. In he model, his is given by V = V 1, + V 2, + η 2 U. Second, regarding he jumps, he only propery ha applies almos surely is he ideniy of he realized jumps in he reurns and sae variables. In paricular, V 1, = µ 1 ( log(x)) 2 1 { log(x)<}, and, if ρ u =, we also have, U = ( log(x)) 2 1 { log(x)<}, under boh measures. In mos empirical opion pricing applicaions, addiional assumpions are invoked when changing measure from P o Q. For example, i is commonly assumed ha he model class is idenical, and affine, under boh measures. This is convenien as affine models offer a grea deal of racabiliy. However, his approach severely resrics he dynamics of he risk premiums. In addiion, such srucure preserving ransformaions (SPTs) impose auxiliary resricions, exending o he model parameers. In paricular, in our affine seing, he SPT assumpion implies ha σ 1, σ 2, σ 3, ρ 1, ρ 2, ρ u, µ 1 and κ 3 are idenical, while he remaining parameers may differ under P and Q. Given he above discussion, i is clear ha daa on he underlying equiy-index values can be helpful in he esimaion of he risk-neural model. Below, we impose he pahwise (almos sure) resricion regarding he spo variance in esimaion. Given he difficuly in recovering spo volailiy jumps from high-frequency daa due boh o esimaion uncerainy and lack of overnigh observaions we do no imposing resricions regarding he pahwise (realized) price and volailiy jump relaionship implied by he risk-neural model (3.1). As deailed above, if one consrains he pricing of risk hrough a SPT from P o Q, addiional informaion from he P dynamics can be impored from he underlying reurn daa during esimaion of he risk-neural dynamics. However, as we documen below, he opion panel is very informaive abou he risk neural dynamics, so we avoid auxiliary resricions ha may induce model misspecificaion. 15

17 4 Esimaion 4.1 Esimaion Approach The developmen of formal ools for parameric inference in he conex of an opion panel is challenging. There are pronounced ime series dependencies in he laen volailiy componens and, poenially, he jump inensiies. A he same ime, sizeable bid-ask spreads influence he observed ransacion prices and quoes for he opions. These measuremen errors for he opions are srongly heerogeneous and correlaed wih he overall reurn variaion. Finally, here are no-arbirage consrains ha, one, a any poin in ime link he individual opion prices across srikes, and, wo, equae he spo diffusive volailiy for he underlying asse (imperfecly observable from high-frequency reurns) wih he volailiy implied by he conemporaneous sae vecor (also exraced wih some degree of saisical error) every ime we observe he opion cross-secion. The ineracions of hese effecs generae complex nonlinear ineracions and dependencies in he sysem which should be accouned for in any pracical assessmen of esimaion precision of qualiy of fi o he opion surface. In his paper, we adap he parameric esimaion and inference approach pu forh in Andersen e al. (213) which is designed o deal wih his exac ype of environmen. I explois in-fill asympoics in he opion cross-secion, i.e., i develops inference ools under he assumpion ha, for a limied se of mauriy daes, opion prices are observed across a broad range of srike prices wih only small gaps beween he exercise prices. The procedure accommodaes ha he opion panel is highly unbalanced, wih a differen se of mauriies and srikes available for each crosssecion, he measuremen errors are heerogeneous and correlaed wih he overall reurn variaion, he explici incorporaion of he no-arbirage consrain linking spo volailiy under he physical and risk-neural measure, and avoids relying on a srucure preserving ransformaion from P o Q ha involves ad hoc resricions on he risk pricing. The approach provides consisen period-by-period esimaes for he sae vecor along wih valid asympoic inference for he sae vecor as well as he model parameers and he model fi o specific regions of he IV surface for any observaion dae. This is feasible only due o he in-fill asympoic scheme for he srikes in he opion cross-secion. 8 A he pracical level, his reflecs he given acual srucure of he panel. The opion quoes are clusered closely across he srike range while he cross-secions are observed weekly. Thus, he well-recognized advanages of 8 Alernaive inference echniques can poenially provide unbiased esimaes for he sae vecor deermining he curren values of he muliple volailiy componens and he jump inensiies, bu hey canno achieve consisency wihin his seing, hus rendering formal inference regarding he sae vecor and IV surface fi on a period-by-period basis infeasible. 16

18 inference via high-frequency observaions apply naurally o he cross-secion, no he ime series, in he given seing. Moreover, he wealh of informaion embedded in he opion cross-secion is, of course, well-recognized. I enables nonparameric exracion of he condiional risk-neural densiy esimaes for he underlying asse reurns across he sequence of mauriies represened by he available enors a a given dae. The evoluion of hese condiional densiies over ime speak o he number of facors, he ineremporal variaion in volailiy and jump inensiies and persisence of he differen characerisics. Given he idenified facors and model parameers, he individual cross-secion idenifies he curren sae vecor. In fac, heoreically, he enire sysem can be idenified and esimaed consisenly from a single opion cross-secion, bu he idenificaion from a single dae is weak and exremely imprecise given he presence of measuremen errors. 9 Finally, we emphasize he advanage of obaining formal diagnosics for model fi. The approach can be used o derive summary measures for wheher we, saisically, suiably accommodae specific regions of he IV surface on a period-by-period basis. Below, we illusrae his via Z-scores ha capure he overall fi for OTM pu and call opions along wih ATM opions a eiher he shor or long mauriies. By comparing hese period-by-period fi across he srike range, we assess he qualiy of fi o he skew, while comparison across he shor and long enors reveal he accommodaion of he IV erm srucure. These indicaions of overall under- or overvaluaion of specific opions explicily accoun for he esimaion uncerainy induced my measuremen errors in he opions, he esimaion uncerainy for he model parameers and period-specific sae vecor and he imprecision in inferring he spo volailiy from he high-frequency reurn daa. 1 Leing n denoe he (equidisan) frequency by which we sample he high-frequency reurns of he underlying asse, our esimaor akes he form, ( {Ŝn } =1,...,T, θ n) = argmin {Z } =1,...,T, θ Θ T 1 =1 N N j=1 ( κ,kj,τ j κ(k j, τ j, Z, θ) ) 2 V n + ζ n ( V (n,m n) V (Z, θ) ) V n ) 2 (4.2), for a deerminisic sequence of nonnegaive numbers {ζ n } (decreasing asympoically o zero, as n diverges) and V (Z, θ) = V 1, + V 2, + η 2 U is he model-implied value of he spo (diffusive) variance. V n V (n,n) and V (n,mn) are nonparameric esimaors of he diffusive inegraed variance consruced from he inraday record of he log-fuures price of he underlying asse. Specifically, 9 On he oher hand, quie reasonable idenificaion can usually be obained from a single year of opion daa along wih he high-frequency reurns on he underlying asse used o enforce he (saisical) equaliy of spo volailiy across he P o Q measures. 1 The massive Mone Carlo sudy in Andersen e al. (213) confirms ha hese es procedures perform well in his ype of seing. 17

19 V (n,mn) explois m n reurns sampled (a frequency n) jus prior o he observaion ime for he opion cross-secion (he end of he rading day). For m n = n, V (n,mn) is he runcaed realized reurn variaion, see, e.g., Mancini (29), which consisenly esimaes he inegraed diffusive variance over a fixed ime period prior o. For m n /n, V (n,mn) is a consisen esimaor of he spo variance a and corresponds o he runcaed realized variaion compued over an (asympoically shrinking) fracion of he day jus prior o he opion quoes. In our implemenaion, we sample every minue over a 6.75 hours rading day, excluding he iniial five minues, resuling in (n,mn) n = 4. We employ m n = 3 for V. Moreover, we explicily accoun for he pronounced inraday volailiy paern. Deails regarding our consrucion of hese esimaors are provided in he appendix. The esimaor (4.2) minimizes he weighed mean squared error in fiing he panel of observed opion implied volailiies, wih a penalizaion erm ha reflecs how much he model-implied spo variance deviaes from a model-free spo variance esimae. The presence of V (n,mn) in he objecive funcion serves as a regularizaion device ha helps idenify he parameer vecor by penalizing values ha imply unreasonable volailiy levels. Moreover, he sandardizaion by V n allows us o weigh opion observaions on high and low volailiy days differenly o improve efficiency by accouning for he fac ha he measuremen errors in he opion prices and he imprecision in esimaing he sae vecor generally rises wih marke volailiy. This represens a deparure from he esimaion procedure in he empirical analysis in Andersen e al. (213). 11 The esimaor (4.2) is derived hrough join opimizaion over he parameers and sae vecor realizaions bu, in pracice, i is mos convenienly performed ieraively, exploiing wo simple seps. For a given parameer vecor, we firs deermine he opimal value of he sae vecor on each day of he sample. Second, for he concenraed objecive funcion, we perform MCMC based opimizaion of he parameer vecor wih a chain of lengh 1,. For his new θ esimae, we hen move back o sep one and ierae he procedure. 4.2 Esimaion Resuls The parameer esimaes for model (3.1) are repored in Table 4. The basic feaures of he volailiy facors, V 1 and V 2, are similar o hose obained via model (??). Noneheless, he size of V 2 has declined significanly relaive o V 1. Moreover, V 2 is less persisen han before wih a half life of abou 18 monhs versus one monh for V 1. In addiion, he volailiy facors exhibi a sronger 11 The raionale for our weighing scheme is apparen from Figure?? in he supplemenary appendix. I demonsraes ha he opion pricing errors are roughly linear in marke volailiy, moivaing our scaling of he errors by he level of volailiy. 18

20 Table 4: Parameer Esimaes for he exended hree facor model Parameer Esimae Sd. Parameer Esimae Sd. Parameer Esimae Sd. ρ σ c v 1.7. µ u c κ κ c σ ρ λ ρ c λ v c µ κ c Noe: Parameer esimaes of he hree-facor model (3.1). The model is esimaed using S&P 5 equiy-index opion daa sampled every Wednesday over he period January 1996-July 21. negaive correlaion wih he diffusive price innovaions. While mos of he jump parameers are no readily comparable o hose in he preceding models, we noe ha he ail decay parameers only change slighly and he overall feaures are quie similar o wha we esablished wih model (??). For example, he negaive and posiive jump inensiies are now abou 3.4 and 2.6 per year, while he jump sizes are abou -4.7% and 2.1%, respecively, represening only minor offseing changes in he jump inensiies and sizes. Likewise, we again find ha he jump inensiy is driven almos exclusively by he ime-varying componen. 12 We also observe ha he c 1 and c+ 1 as well as he c 2 and c+ 2 coefficiens are pairwise saisically disinc, indicaing a differen degree of imevariaion in he lef and righ jump ail, consisen wih he nonparameric evidence in Bollerslev and Todorov (211b). Furher, he sae variable U is highly persisen and, as discussed in deail laer, is he primary deerminan behind he ime variaion of he lef jump ail. Finally, he parameer ρ 3 canno be esimaed wih precision, implying ha he degree of independen jump variaion in U is hard o idenify from he opion panel alone. The exended hree-facor model (3.1) improves he fi o he opion surface subsanially relaive o model (??). The RMSE for he opion-implied volailiies drops by close o 15% and now equals 1.75%. Even more srikingly, he Z-scores in Figure?? porray a se of saisics ha largely reside wihin he 95% confidence bands. The abulaion of opion pricing violaions a he 1% level across he six srike-mauriy caegories in he Supplemenary Appendix corroboraes he poin. 12 As before, he esimaes of c were insignifican and fixed a zero. 19