Inferring volatility dynamics and risk premia from the S&P 500 and VIX markets

Transcription

1 Inferring volailiy dynamics and risk premia from he S&P 500 and VIX markes Chris Bardge Elise Gourier Markus Leippold November 2016 Absrac This paper shows ha he VIX marke conains informaion on he variance of he S&P 500 reurns, which is no already spanned by he S&P 500 marke. We esimae a flexible affine model based on a join ime series of underlying indexes and opion prices on boh markes. We find ha including VIX opion prices in he model esimaion allows beer idenificaion of he parameers driving he risk-neural condiional disribuions and erm srucure of volailiy, hereby enhancing he esimaion of he variance risk premium. We gain new insighs on he properies of he premium s erm srucure and show how hey can be used o form rading signals. Finally, we show ha our premium, used ogeher wih a measure of is erm srucure, has beer predicive power on S&P 500 reurns compared o he usual model-free premium. Keywords: S&P 500 and VIX join modeling, volailiy dynamics, paricle filer, variance risk premium. JEL Codes: G12, G13, C58. The auhors hank Yacine Aï-Sahalia, Torben Andersen, Tim Bollerslev, Peer Chrisoffersen, Jérôme Deemple, Garland Durham, Damir Filipović, Andras Fulop, Michael Johannes, Loriano Mancini, Chay Ornhanalai, Chris Rogers, Olivier Scaille, Ronnie Sircar, Josef Teichmann, Fabio Trojani, Anders Trolle and Alexandre Ziegler for helpful commens. Useful feedback was received from seminar paricipans a Universiy of Zürich, ETH, Universiy of Aarhus, HEC Monreal, Queen Mary Universiy, Ken Business School, Toulouse School of Economics as well as a he Swiss Finance Insiue Gerzensee workshop, he Bachelier Finance Conference, EFA 2013, he 2014 Conference of he Swiss Sociey for Financial Marke Research, he 5h Inernaional Conference of he ERCIM WG on Compuing & Saisics, and he 2015 French Economerics Conference. Financial suppor from he Swiss Finance Insiue SFI), Bank Vonobel, he Swiss Naional Science Foundaion and he Naional Cener of Compeence in Research Financial Valuaion and Risk Managemen is graefully acknowledged. Universiy of Zurich and Swiss Finance Insiue SFI), Plaensrasse 14, 8032 Zürich, Swizerland; el: +41) ; chris.bardge@bf.uzh.ch. Queen Mary Universiy of London and Cener for Economic Policy Research CEPR), Mile End Road, London E1 4NS, UK; e.gourier@qmul.ac.uk. Universiy of Zurich and Swiss Finance Insiue SFI), Plaensrasse 14, 8032 Zürich, Swizerland; el: +41) ; markus.leippold@bf.uzh.ch.

2 1 Inroducion Inroduced by he CBOE in 1993, he VIX index non-paramerically approximaes he expeced fuure realized volailiy of he S&P 500 reurns over he nex 30 days. Opions on he VIX sared rading in 2006 and, as of oday, represen a much larger marke han VIX fuures. By definiion, he VIX index, VIX opions, and S&P 500 opions are direcly linked o he S&P 500 index and all provide valuable informaion on he S&P 500 reurns dynamics. However, o he bes of our knowledge, here has been very lile effor dedicaed o comparing he informaion hese daases conain on he disribuion of he S&P 500 reurns and on he rajecory of heir variance process. In his paper we aim o fill his gap and sudy he added informaion conen of he VIX opion marke compared o he S&P 500 marke. Our main conribuion o he empirical opion pricing lieraure is o show ha VIX opions conain valuable informaion on he dynamic properies of S&P 500 reurns, which is no spanned by S&P 500 opions. We draw his conclusion from a parameric approach, using a parsimonious and flexible affine model for reurns. Our resuls are backed by various in- and ou-of-sample ess as well as an in-deph analysis of he implied variance risk premium VRP). We argue ha VIX opions allow for an enhanced represenaion of he VRP and of is erm srucure, and show ha he resuling VRP can be used o form rading signals and improve predicions of S&P 500 reurns. Joinly analyzing he dynamic properies and informaion conen of he VIX and S&P 500 opion markes is a challenge. No only do we need a model ha is flexible enough o simulaneously accommodae he sylized facs of boh markes over ime, bu he empirical analysis of such highly nonlinear daa poses a significan compuaional hurdle. We develop a ime-consisen esimaion procedure ha permis us o exrac informaion from a large and unbalanced panel of daa and esimae he rajecories of he unobserved volailiy of he S&P 500 reurns. This mehodology goes well beyond a simple calibraion exercise as i makes i possible o reconcile ime series daa on he S&P 500 and VIX derivaives markes and consisenly mach he join evoluion of prices over ime. We model he S&P 500 reurns using an affine jump-diffusion specificaion ha belongs o he class of Duffie, Pan, and Singleon 2000). This specificaion feaures wo facors driving he variance 1

3 process, and an addiional facor driving he jump inensiy. Is affine srucure allows us o price S&P 500 and VIX derivaives in semi-closed form, which is essenial o analyze he reurns and volailiy dynamics using a large daase of opions. I also enables us o derive he VRP in closedform, and o conduc a horough analysis of is dynamic behavior and erm srucure. We find ha VIX opions conain informaion on he dynamics of he S&P 500 reurns and heir variance which is no spanned by S&P 500 opions, disregarding he sae of he economy. More specifically, in ime of marke calm, VIX opions do no bring any value in esimaing he curren sae of laen facors. However, hey allow beer idenificaion of he parameers of he model, hereby providing informaion on he condiional disribuions of he underlying reurns and heir variance. This ranslaes ino a beer pricing of VIX opions, which are no well priced when no included in he esimaion daase. This observaion holds boh in- and ou-of-sample, and is herefore no he resul of over-fiing. I also holds in imes of marke urmoil, bu hen VIX opions bring value in idenifying he curren saes of laen processes as well. Furhermore, we show ha adding VIX opions o he esimaion allow a beer represenaion of he erm srucure of variance. We synhesize he VIX index from S&P 500 opions for mauriies from 2 o 6 monhs. Our model, when esimaed o a daase which does no include VIX opions, yields RMSEs which are more han wice as large as he ones obained when VIX opions are included in he esimaion daase. Our resuls have considerable impac in erms of pricing and risk managemen, which rely heavily on an accurae esimaion of he condiional disribuions of he underlying risk facors over differen ime horizons. A by-produc of our esimaion is he variance risk premium VRP), which represens he compensaion invesors expec o receive for bearing he risk coming from sochasic flucuaions in he variance of he reurns over a given horizon. In oher words, i is he expeced payoff of a variance swap. By definiion, i depends on he condiional expecaion of he variance of S&P 500 reurns. Due o he affine srucure of our model, he VRP is available in closed-form, which enables us o address hree highly debaed quesions in he recen lieraure: Wha are he main componens driving he VRP? Can he VRP be used o form rading signals? Does he VRP have predicive power on S&P 500 reurns? 2

4 We find ha he VRP is very sensiive o jumps in he reurns and heir variance, in paricular when he invesmen horizon is shor. Inuiively, his is jusified by he fac ha a large movemen in he variance process has an immediae negaive impac on he payoff of a shor-erm variance swap. The wo variance facors are shown o have differen effecs on he VRP. The firs facor reacs swifly o changing marke condiions and capures mos of he sudden variance flucuaions, especially during marke urmoil. As such, is impac on he VRP dominaes for shor-erm invesmens and during urmoil periods. The second facor is more persisen and capures mid- o long-erm rends of he reurn variance. Is impac on he VRP is mos imporan during calm marke periods and for mido long-erm invesmens. Our resuls on he erm srucure of VRP complemens recen findings of Gruber, Tebaldi, and Trojani 2015). In line wih he laer auhors, we find ha ha he VRP has a downward sloping erm srucure in imes of marke calm bu ha his is no longer rue during high volailiy imes. During hese periods, he convexiy of he VRP erm srucure increases, i.e., he VRP becomes more negaive when increasing he invesmen s mauriy up o 3 o 4 monhs, and hen increases. While he usual rading sraegy o reap he VRP is o buy long-erm variance conracs for example variance swaps, see Filipović, Gourier, and Mancini 2016)) and sell shor-erm conracs, we show ha he change in he erm srucure of VRP can be inerpreed as a rading signal. Our proposed sraegy achieves a Sharpe raio of 1.02 over he urbulen period from 2006 o 2010, agains 0.01 for he usual sraegy. Finally, we examine he predicive power of our model-implied VRP on S&P 500 reurns. We find ha he erm srucure of he VRP has predicive power on S&P 500 reurns as well. Indeed, adding a measure of skewness of he VRP erm srucure or, equivalenly, of convexiy), allows reaching an R 2 of 0.06 when he esimaion daase does no conain VIX opions i is composed of S&P 500 opions and of he underlying S&P 500 and VIX indexes). This R 2 is no saisically differen from zero wihou he convexiy measure. Furhermore, he R 2 increases above 0.12 when including VIX opions in he esimaion daase. As we adop a parameric approach, our resuls are backed up by an exensive model specificaion analysis. We examine differen nesed models o invesigae he role of he various feaures in ex- 3

5 plaining opion prices, he risk-neural disribuions of reurns, and hose of he variance process. Of course, any parameric approach is bound o suffer, o a cerain exen, from model misspecificaion. Based on likelihood crieria as well as analyses of he in- and ou-of sample pricing errors, we show ha he full specificaion of our model is needed o represen he underlying indices as well as he opions on boh markes. We address he compuaional challenge of joinly esimaing a model o wo liquid opion markes by designing an opion pricing algorihm and a paricle filer, which are ailored o our problem and model specificaion. Esimaing he dynamics of he S&P 500 reurns from an exremely large daase of opions on he wo markes and for a long ime series requires compuaionally efficien echniques ha can easily deal wih he feaures of he model, in paricular he sae-dependen jumps. To achieve his goal, we exend he Fourier Cosine mehod inroduced by Fang and Ooserlee 2008) for S&P 500 opions o price VIX opions and adap he Auxiliary Paricle Filer of Pi and Shephard 1999) o esimae he rajecories of unobservable processes and jumps. Accordingly, we provide an exensive oolki for inference and diagnosics of affine opion pricing models given index and opion daa from boh he S&P 500 and VIX markes. Paricle filering echniques and more generally Sequenial Mone Carlo mehods have recenly increased in populariy and have been used o esimae models, bu mos endeavors using his ool resric heir opions daase o near a-he-money opions and as far as we know, none have used S&P 500 and VIX derivaives joinly. Our work is relaed o several recen papers which aim o reconcile he cross-secional informaion of he S&P 500 and he VIX derivaives markes by modeling hem joinly. Gaheral 2008) poined ou firs ha even hough he Heson model performs fairly well a pricing S&P 500 opions, i fails o price VIX opions. In fac, modeling he insananeous volailiy as a square roo process leads o a VIX smile decreasing wih moneyness, which is he opposie of wha is observed in pracice. Among he recen papers ha have aemped o simulaneously reproduce he volailiy smiles of S&P 500 and VIX opions are Chung, Tsai, Wang, and Weng 2011), Con and Kokholm 2013), Papanicolaou and Sircar 2013), and Bayer, Gaheral, and Karlsmark 2013). We build on his lieraure by considering exensions of he Heson model ha remain wihin he affine framework, bu add more flexibiliy o he specificaions used in he above menioned papers. We use a special 4

6 case of he general affine framework developed by Duffie, Pan, and Singleon 2000) ha includes as sub-cases he usual exensions of he Heson model encounered in he lieraure, for example Baes 2000b), Eraker 2004), and Sepp 2008a). 1 In relaed work, Song and Xiu 2015) use a model ha is similar o ours bu wih a differen focus, and esimae marginal densiies and pricing kernels of he marke reurns and VIX. We also build on a lieraure which sudies he dynamic properies of variance risk premia. Amengual 2008) uses S&P 500 opions and variance swaps o infer he erm srucure of variance risk premia prior o he financial crisis. He finds a downward-sloping erm srucure of variance risk premia, which is confirmed by laer sudies by Andries, Eisenbach, Schmalz, and Wang 2015), solely based on S&P 500 opions. Gruber, Tebaldi, and Trojani 2015) are he firs o differeniae beween periods of low and high volailiy, and find ha he slope of he VRP erm srucure swiches sign in periods of disress. Our resuls exend heirs as we show ha he erm srucure of VRP is no longer monoonic in imes of high volailiy: i has negaive slope up o hree o four monhs, and hen posiive slope. On a relaed noe, Dew-Becker, Giglio, Le, and Rodriguez 2016) show ha on average, invesors do no price news abou fuure volailiy and are only willing o hedge agains flucuaions in shor-erm realized volailiy. We argue ha he aiude of invesors owards long-erm volailiy changes over ime, and depends on marke condiions. Finally, our work enriches he lieraure on ime-consisen esimaion mehods. These mehods have been previously used o calibrae models o index reurns and opions. See, e.g., Baes 2000a), Pan 2002), Eraker 2004), Broadie, Chernov, and Johannes 2007), Chrisoffersen, Jacobs, and Mimouni 2010), Johannes, Polson, and Sroud 2009) and Duan and Yeh 2011). However, as underlined in Ferriani and Pasorello 2012), mos papers filering informaion from opion prices rely on one opion per day or a limied se of opions. Limiing he amoun of daa resuls in a compuaionally less inensive empirical exercise, bu i ignores a large par of he informaion presen in he markes. In conras, in our paricle filer esimaion we fully exploi he richness of our daase. Furhermore, we noe ha mos, if no all papers ha consider S&P 500 and VIX opions in heir calibraion exercise 1 Some sudies are going in he direcion of non-affine models e.g., Jones 2003), Aï-Sahalia and Kimmel 2007), Chrisoffersen, Jacobs, and Mimouni 2010), Ferriani and Pasorello 2012), Durham 2013), Kaeck and Alexander 2012)). However, racabiliy remains an issue ha is of crucial imporance when i comes o calibraing a model o a long ime series conaining hundreds of opions each day. 5

7 have resriced heir analysis o a saic one-day esimaion. The resuling parameers migh exhibi large variaions when calibraing he model o differen daes and herefore canno be used o infer ime series properies of reurns and risk premia. 2 This paper is organized as follows. In Secion 2, we inroduce he hree-facor affine jump diffusion framework used laer in he esimaion. We describe he risk premium specificaion and derive he expression for he VIX squared as well as he pricing formula for VIX and S&P 500 opions. In Secion 3, we describe our daase. In Secion 4, we deail our ime series consisen esimaion mehod. In Secion 5, we discuss our esimaion resuls and model specificaion analysis. Secion 6 provides a horough analysis of he VRP, he properies of is erm srucure and predicive power on fuure S&P 500 reurns. Secion 8 concludes. 2 Theoreical framework We firs presen our modeling framework. Our model is novel and able o represen imporan sylized facs of S&P 500 reurns, which have been recenly highlighed in he lieraure. In paricular, i includes a sae-of-he-ar represenaion of he jumps, inspired from Andersen, Fusari, and Todorov 2015) and Amengual and Xiu 2015), which allows beer capuring he sochasic skewness of reurns and of heir variance. Despie is flexibiliy, i is parsimonious and belongs o he affine class of models, which ensures racabiliy. 2.1 Model specificaion Le Ω, F, {F } 0, P) be a filered probabiliy space saisfying he usual assumpions, where P denoes he hisorical measure. We consider a risk-neural measure Q equivalen o P. Le F ) 0 be he forward price of he S&P 500 index and Y = Y ) 0 = logf )) 0 he reurns. The dynamics 2 See, e.g., Lindsröm, Sröjby, Brodén, Wikorsson, and Hols 2008). 6

8 of Y under Q are specified by dy = µv, m, u )d + v dw Y + dj Y +) dv = κ v m v )d + σ v v dw v + dj v+) + dj Y ), 1) + dj v ), 2) dm = κ m θ m m )d + σ m m dw m, 3) where W Y, W v and W m are sandard Brownian moions. The processes W Y and W v are correlaed wih coefficien ρ Y,v. All oher Brownian moions are muually independen. The process v = v ) 0 is he diffusive componen of he variance of S&P 500 reurns. The second variance facor m = m ) 0 represens a sochasic level around which v revers. 3 We refer o i as cenral endency. The wo processes v and m are insananeously uncorrelaed and only inerac via he drif erm of v. The processes J Y +) processes N Y +), J Y ), J v+) and J v ) are finie aciviy jump processes driven by he poin and N Y ). The process N Y ) resp. N Y +) ) couns negaive resp. posiive) jumps in he reurns. The jump processes are defined by N Y +) J Y +) = i=1 Z Y +) i N Y ), J Y ) = i=1 Z Y ) i N Y ), J v+) = i=1 Z v+) i N Y +), J v ) = i=1 Z v ) i. 4) where Z Y +) i, Z Y ) i, Z v+) i and Z v ) i are he random jump sizes. As suggesed by he price pahs of he S&P 500 and VIX index, large negaive res. posiive) movemens in equiy reurns and large posiive resp. negaive) movemens in he variance are likely o occur a he same ime. We herefore choose, in line wih he lieraure, he same poin processes N Y +) and N Y ) o generae jumps in he asse reurns and variance process v. The leverage effec is driven by he correlaion beween W Y and W v as well as he possibiliy of simulaneous jumps in he reurns and variance. While i is sandard o model posiive jumps in he reurns, accouning for negaive jumps has less been done. 3 I has already been shown ha a leas wo facors are needed o provide an accurae descripion of he volailiy dynamics see, e.g., Baes 2000b), Andersen, Benzoni, and Lund 2002), Alizadeh, Brand, and Diebold 2002), Chernov, Gallan, Ghysels, and Tauchen 2003), Chrisoffersen, Heson, and Jacobs 2009), Egloff, Leippold, and Wu 2010), Todorov 2010), Kaeck and Alexander 2012), Baes 2012), Johnson 2012), Mencía and Senana 2013), Huang and Shaliasovich 2015) and Branger, Krafschik, and V olker 2016)). 7

9 Amengual and Xiu 2015) show ha negaive jumps in volailiy do occur, and are usually riggered by macroeconomic announcemens. We assume ha he jump inensiies depend linearly on levels of he diffusive laen processes v, m and u. 4 The inensiy of posiive jumps in reurns is denoed by λ +), and he inensiy of negaive jumps by λ ) : λ +) v, m ) = λ +) 0 + λ +) X ; λ ) v, m, u ) = λ ) 0 + λ ) X, 5) where X denoes he column) vecor of diffusive laen processes X = v, m, u ), λ +) = λ +) 1, λ +) 2, 0) and λ ) = λ ) 1, λ ) 2, λ ) 3 ). The process u drives he inensiy of negaive jumps in he reurns join wih posiive jumps in he variance), as in Andersen, Fusari, and Todorov 2015). I has he following dynamics: du = κ u θ u u )d + σ u u dw u, 6) wih W u independen of he oher Brownian moions. Inuiively, u allows o beer represen he sochasic skewness of he reurn process. Andersen, Fusari, and Todorov 2015) find ha he effec of he process u on he inensiy of posiive jumps in reurns and on he diffusive variance is insignifican, herefore we do no incorporae i in our model. We assume ha he random jump sizes are independen and idenically disribued. The jump sizes in he reurns and volailiy are assumed o be exponenially disribued wih respecive means µ +) Y, µ ) Y, ν+) v and ν ) v. Le us define Z i = Z Y +) characerized by heir join Laplace ransform i, Z Y ) i, Z v+) i, Z Y ) i ), i N. The jump sizes are θ Z φ) = θ Z1 φ +) Y, φ ) Y, φ+) v, φ ) v ) = E Q [expφ Z 1 )], φ C 3. 7) 4 The specificaion of jumps is of imporance. Todorov 2010), Todorov and Tauchen 2011) and Jacod and Todorov 2010) find sriking evidence for co-jumps in S&P 500 reurns and in he VIX. See also Eraker 2004), Broadie, Chernov, and Johannes 2007), Con and Kokholm 2013). Baes 1996), Pan 2002) and Eraker 2004) argue in favor of using sae-dependen jumps in reurns, which is inuiively appealing, as jumps end o occur more frequenly when volailiy increases. Using variance swaps, Aï-Sahalia, Karaman, and Mancini 2012) find ha he sae dependen inensiy of jumps is a desirable model feaure. 8

10 The drif of he reurns process can be wrien accordingly as: µv, m, u ) = λ ) v, m, u )θ Z 0, 1, 0, 0) 1) λ +) v, m )θ Z 1, 0, 0, 0) 1) 1 2 v 8) In our model, he diffusive variance of reurns can in heory reach zero wih posiive probabiliy as well as become negaive because of he negaive jumps in v. While his is cerainly a drawback, he empirical lieraure has highlighed he need for negaive jumps in volailiy and shown ha even wih a sandard Heson model wih posiive jumps only, imposing he Feller condiion pus resricions on he model which do no allow a saisfacory represenaion of he daa; see, e.g., Song and Xiu 2015). To ackle his issue, we verify in he empirical par of he paper ha he esimaed rajecory of he process v never crosses he zero boundary. This model specificaion implicily defines he dynamics for he VIX. To derive is expression wihin our framework, we use he definiion of he VIX as a finie sum of call and pu prices ha converges [ ] o he inegral VIX 2 = 2 +τ τ EQ df u F dln F u u), where τ is 30 days in annual erms. Proposiion 2.1. Under he model specificaion given in Equaions 1)-7), he VIX squared a ime can be wrien as an affine deerminisic funcion of v and m : VIX 2 = 1 τ EQ +τ v u du + 2 N Y v +τ i=n Y v e ZY i 1 Z Y i ), 9) = α VIX 2v + β VIX 2m + γ VIX 2u + δ VIX 2, 10) where he coefficiens α VIX 2, β VIX 2, γ VIX 2 and δ VIX 2 are known in closed form and provided in Appendix A. 2.2 Risk premium specificaion We specify he change of measure from he pricing o he hisorical measure so ha he model dynamics keep he same srucure under P. We separae he oal equiy risk premium γ ino a diffusive conribuion, which is proporional o he variance level and represens he compensaion 9

11 for he diffusive price risk, and a jump conribuion reflecing he compensaion for jump risk: ) ) γ = η Y v +λ +) v, m ) θz1, P 0, 0, 0) θ Z 1, 0, 0, 0) +λ ) v, m, u ) θz0, P 1, 0, 0) θ Z 0, 1, 0, 0), 11) where θ P Z denoes he join Laplace ransform of jump sizes under he hisorical measure P. We follow Pan 2002) and Eraker 2004) and assume ha he inensiy of jumps is he same under Q and P. 5 However, we allow he mean of he jump sizes in reurns o be differen under Q and P. Similarly, he volailiy risk premium on he wo volailiy facors v and m decomposes ino a diffusive componen and a jump componen. The diffusive variance risk premium in v is proporional o he curren level of variance, wih coefficien of proporionaliy given by η v = κ v κ P v. The same applies o he cenral endency m, for which he coefficien is defined as η m = κ m κ P m. For he jump par of he volailiy risk premium, we allow he mean jump sizes ν +) v and Q. and ν ) v o be differen under P In line wih Andersen, Fusari, and Todorov 2015), we find in he empirical par ha he rajecory of he facor u is difficul o esimae. Therefore, o no add unnecessary complexiy o he model, we assume ha i does no carry any risk premium. 2.3 Derivaives pricing Wihin he class of affine models, opion pricing is mos efficienly performed using Fourier inversion echniques. As a saring poin, we need he characerisic funcion of he underlying processes. Due o he affine propery of he VIX square in Proposiion 2.1, we have he following resul: Proposiion 2.2. In he wo-facor sochasic volailiy model wih jumps defined by equaions 1)- 7), he Laplace ransforms of he VIX square and he S&P 500 reurns are exponenial affine in he 5 Pan 2002) argues ha inroducing differen inensiies of jumps under he hisorical and pricing measure inroduces a jump-iming risk premium ha is very difficul o disenangle from he mean jump risk premium. Our assumpion arificially incorporaes he jump-iming risk premium ino he mean jump size risk premium. 10

12 curren values of he facor processes v, m and u: Ψ VIX 2 T, x; ω) := E Q Ψ YT, y, x; ω) := E Q [ [ e ωvix2 T e ωy T ] X = x = e αt )+BT ) x, ] y = y, X = x = e α Y T )+β Y T )y+b Y T ) x, where B = β, γ, δ) and B Y = γ Y, δ Y, ξ Y ) are funcions defined on [0, T ] by he ODEs given in Appendix B. The parameer ω belongs o a subse of C where he above expecaions are finie. Pricing opions on he VIX poses echnical difficulies ha are no encounered when pricing equiy opions. Given a call opion wih srike K and mauriy T on he VIX a ime = 0, we need o calculae CVIX 0, K, T ) = e rt v K) + f VIX 2 v)dv, 12) T 0 where f VIX 2 T is he Q-densiy of he VIX square a ime = T. The square roo appearing in he inegral as par of he payoff in 12) prevens us from using he Fas Fourier Transform of Carr and Madan 1999). We would need he log of he VIX o be affine, which is incompaible wih affine models for log-reurns. However, his problem can be circumvened. Fang and Ooserlee 2008) inroduce he Fourier cosine expansion o price index opions on he S&P 500. We exend heir mehod o ackle he pricing of VIX opions. Our approach o pricing VIX opions is comparable o he inversion performed by Sepp 2008a) and Song and Xiu 2015), bu i is more parsimonious in he number of compuaional parameers. Proposiion 2.3. Consider a European-syle coningen claim on he VIX index wih mauriy T and payoff u VIX VIX 2 ) = VIX 2 K) +. Given an inerval [a VIX, b VIX ] for he suppor of he VIX 2 T v 0,m 0 densiy, he price P VIX 0, VIX 0 ) a ime = 0 0 of he coningen claim is N 1 P VIX 0, VIX 0 )= e rt 0) A VIX2 n Un VIX2, 13) n=0 11

13 where he prime superscrip in he sum The erms in he sum are defined by A VIX2 n = U VIX2 n = 2 Re b VIX a VIX bvix a VIX u VIX v) cos {Ψ VIX 2T 0, v 0, m 0 ; means ha he firs erm A VIX2 0 U VIX2 0 is divided by 2. inπ ) exp ia VIX nπ )}, 14) b VIX a VIX b VIX a VIX nπ v a ) VIX dv. 15) b VIX a VIX The coefficien A VIX2 n in Appendix C. is compued using Proposiion 2.2 and U VIX2 n is known in closed form and given 3 Daa and preliminary analysis In his secion, we describe our daa and poin ou some imporan characerisics of VIX opions. 3.1 Daa descripion Opions on he VIX were inroduced in Our sample period is from March 1, 2006 o Ocober 29, The opion daa consis of he daily closing prices of European opions on he S&P 500 and VIX, obained from OpionMerics. This ime series includes boh periods of calm and periods of crisis wih exreme evens. Therefore, i provides an ideal es bed for our candidae models. Boh he S&P 500 and VIX opions daases are reaed following he lieraure, see e.g. Aï-Sahalia and Lo 1998). We only consider opions wih mauriies beween one week and one year and delee opions quoes ha are no raded on a given dae. Then, we infer from highly liquid opions he fuures price using he a-he-money ATM) pu-call pariy. This avoids wo issues: Making predicions on fuure dividends, and using fuures closing prices which are no synchronized wih he opion closing prices. Hence, we consider ha he underlying of he opions is he index fuures and no he index iself. We only work wih liquid ou-of-he-money OTM) opions for he S&P 500 marke and only wih liquid call opions for he VIX marke. If he VIX in-he-money ITM) call is no liquid, we use he pu call pariy o infer a liquid VIX ITM call from a more liquid VIX OTM 12

14 pu. Finally, implied volailiies are compued considering fuures prices as underlying. 6 These adjusmens leave a oal of 383,286 OTM S&P 500 and 43,775 call opions on he VIX, wih a daily average of 327 S&P 500 opions and 37 VIX opions. The number of S&P 500 VIX) opions in our daase on a given dae increases wih ime, wih around 170 5) opions a he beginning of he daase and around ) opions a he end. A he beginning of he sample, here are one or wo shor mauriies less han six monhs) available for VIX opions and around six mauriies for S&P 500 opions, wih approximaely 40 S&P 500 opions per mauriy slice. A he end of he sample, he VIX opions have around five shor mauriies wih a bi more han 10 opions rading per mauriy. For S&P 500 opions, around en mauriies are available per day wih around 60 opions for one-monh mauriies and 40 opions for he one-year slice. The low number of VIX opions compared o he number of S&P 500 opions comes from he fac ha VIX opions only sared rading in A he end of our sample, he oal VIX opions volume per day is abou one-half he oal volume of S&P 500 opions raded. 3.2 Descripive saisics Table 1 presens he firs four sample momens of he S&P 500 fuures reurns and VIX index levels, over wo differen periods of ime. The firs period sars in March 2006 and ends in February 2009, i.e., i spans he pre-crisis period as well as he beginning of he crisis. The second period begins in March 2009 and lass unil Ocober For our esimaion, hese wo periods serve as he in-sample and ou-of-sample periods. The S&P 500 log-reurns exhibi a high kurosis, especially during he in-sample period, suggesing he presence of rare and large movemens. In he in-sample period, heir skewness is slighly posiive, bu becomes negaive in he ou-of-sample period, due o he subsanial losses made during he financial crisis. The VIX index exhibis a large posiive skewness and kurosis in he in-sample period. However, in he ou-of-sample period, boh skewness and kurosis decrease significanly 6 We remark ha VIX opion prices do no saisfy no-arbirage relaions wih respec o he VIX index, bu raher wih respec o he VIX fuures value. A VIX call opion a ime wih mauriy T is an opion on he volailiy for he ime inerval [T, T + 30d], where 30d sands for 30 days. The value VIX a ime is relaed o he volailiy on he ime inerval [, + 30d], which migh no overlap a all wih [T, T + 30d]. 13

15 whereas he mean increases by 45%, indicaing ha he values of he index are of larger overall magniude bu wih less exreme values. [Table 1 abou here.] Panel A of Figure 1 displays he join evoluion of he S&P 500 and he VIX index from 2006 o The S&P 500 reurns and he VIX daily incremens are highly negaively correlaed wih a correlaion coefficien of over his period), which explains he populariy of VIX conracs for hedging par of he equiy risk of a porfolio. Panel B represens he expeced one-monh forward log-reurns of he S&P 500 from March 1s, 2006 o Ocober 29h, 2010 as implied by prices of S&P 500 opions and calculaed following he mehod of Bakshi, Kapadia, and Madan 2003). The expeced forward reurns illusrae he variey of marke siuaions covered by our ime series. They were almos consan unil he end of 2007, equal o a posiive value and hus indicaing ha marke paricipans were expecing a sable income from invesing in he index. From he end of 2007, hey exhibi more variaion and evenually urn negaive. Following he bail-ou of Lehman Brohers in Sepember 2008, he expeced forward reurns drop below -1.5%. Then, hey gradually come back and become sable mid-2009 around a slighly negaive level close o -0.2%. In 2010, he sudden increase in he VIX index coincides wih a furher sudden drop of he expeced forward reurns falling o almos -0.5%. [Figure 1 abou here.] Even hough he S&P 500 and VIX markes are closely relaed, we emphasize ha opions on he VIX and S&P 500 differ subsanially in heir characerisics and in he informaion ha hey conain on he underlying S&P 500 reurns and variances. Firs, S&P 500 and VIX derivaives wih he same mauriy conain informaion on he S&P 500 over differen ime periods. While an S&P 500 opion wih mauriy T conains informaion abou he fuure S&P 500 index level a ime T and herefore abou he S&P 500 volailiy up o T, a VIX opion wih mauriy T embeds informaion abou he VIX a ime T and herefore abou he S&P 500 volailiy beween T and T + 30 days. Second, he wo ypes of opions also differ in heir conens. While S&P 500 opions, assuming a coninuous range of raded srikes, characerize he condiional densiy of fuure S&P 500 reurns, i only provides a poin esimae, a each raded mauriy, of he condiional variance of reurns. In 14

16 urn, VIX opions characerize he whole condiional densiy of fuure VIX levels, and as such are likely o conain much more informaion on he fuure densiy of S&P 500 reurn variance. Panels C and D of Figure 1 display he S&P 500 and VIX smiles on May 10, The implied volailiies IVs) are compued using he sandard Black Scholes formula. The VIX IVs are in general subsanially higher han he S&P 500 IVs. They range in our sample from 34% o 216% wih an average of 80% whereas he S&P 500 IVs range from 6% o 162% wih an average of 26%. The implied volailiies of S&P 500 opions are in general decreasing wih moneyness, which highlighs he expensiveness of ou-of-he money pu opions on he S&P 500. As hese opions provide invesors wih proecion agains large downward movemens in reurns, he negaive skewness of he volailiy smile reflecs heir risk aversion owards such movemens. Due o he leverage effec, negaive changes in reurns are srongly relaed o increases in volailiy, which ou-of-he-money VIX call opions can hedge. This explains why VIX implied volailiies end o be posiively skewed. A relaed quaniy is he pu call rading raio. Almos wice as many pus as calls are raded daily in he S&P 500 opions marke, bu he siuaion is reversed in he VIX marke, where he amoun of calls raded daily is almos double ha of he pus. In fac, we can observe in Panels C and D of Figure 1 ha he log-moneynesses raded for S&P 500 opions are mosly negaive which corresponds o OTM pu opions) and ofen posiive for VIX opions OTM calls). 4 Esimaion mehodology Before we bring our model o he ime series of daa, we carry ou a join calibraion exercise using he cross-secion of S&P 500 and VIX opions on a paricular dae. This exercise gives us some guidance for model design and allows us o reduce he se of models o be esimaed on a ime series of opions daa. If a model is no flexible enough o joinly reproduce he implied volailiy paerns of boh markes on a single dae, he Q dynamics of he model is no sufficienly rich o accuraely price boh he S&P 500 and VIX derivaives joinly, and we can safely discard his model from furher consideraion. Our mehodology and deailed resuls are repored in Appendix E and illusraed in Figure 2. We consider wo sub-specificaions of our full model SVJ3): i) m and u 15

17 are consan SVJ), ii) m is sochasic bu u is consan SVJ2). In he full SVJ3 specificaion, we impose θ u o 1 and λ ) 0 = 0 o improve idenificaion. We find ha irrespecive of he day, he SVJ specificaion performs as well as he SVJ2 and SVJ3 specificaions on he S&P 500 opions marke, all fiing he daa very well. In conras, we find ha here are daes when he SVJ model sruggles o fi he VIX IVs in addiion o he S&P 500 IVs, whereas he SVJ2 and SVJ3 models saisfacorily fi boh. 7 Therefore, we conclude from our calibraion exercise ha we can discard he Heson model from furher analysis and ha jumps in he volailiy are necessary. [Figure 2 abou here.] Daily calibraion is essenially a muliple curve fiing exercise, which maches a model o riskneural disribuions implied by opion prices a differen mauriies. Some of he parameers we ge from daily calibraions are unsable and vary subsanially from one day o he nex. 8 To achieve a more robus esimaion, consisen wih he whole ime series of in-sample daa, we choose a mehodology based on paricle filering. A paricle filer uses a ime series of observable marke daa, called measuremens, o esimae he condiional densiies of unobserved laen processes such as he volailiy and jump processes a every poin in ime during he esimaion period. I can be combined wih maximum likelihood esimaion for parameer esimaion and sandard error calculaions. Using a ime series of S&P 500 and VIX indexes and opions, we esimae boh he P- and Q-dynamics of he model o obain a se of model parameers ha joinly prices spo and opions in boh markes consisenly over ime. The esimaion is performed over he in-sample period. The ou-of-sample analysis is conduced by seing he parameers equal o heir in-sample esimaes and running he filer on he subsequen period. Before inroducing he specific filer used, we specify he discreized model and he measuremen errors. 7 Our findings are consisen wih Gaheral 2008), who shows ha he Heson model is incapable of reproducing he posiive skew in VIX IVs, and wih Sepp 2008a,b), who finds ha incorporaing posiive jumps in he volailiy dynamics ino he Heson model removes his shorcoming. 8 Parameers obained when calibraing o daily opions prices are no sable over ime, as explained in Broadie, Chernov, and Johannes 2007) and Lindsröm, Sröjby, Brodén, Wikorsson, and Hols 2008). 16

18 4.1 Discreized model and specificaion of errors We discreize he coninuous-ime model on a uniform ime grid composed of M + 1 poins { 0 = 0, 1 =,..., = k,..., M = M }, for some M N. Since we use daily daa, corresponds o one day. In discree ime, he model evolves under P as follows: Y = [ λ ) v, m, u )θ P Z0, 1, 0, 0) 1) λ +) v, m )θ P Z1, 0, 0, 0) 1) 1 2 v + γ ] v = κ P v + v W Y,P κv + Z Y +),P N +) ) κ P m v + σ v v W v,p + Z v ),P N +) v + Z Y ),P N ), 16) + Z v+),p N ), 17) m = κ P mθ P m m ) + σ m m W m,p, 18) u = κ u θ u u ) + σ u m W u,p, 19) where he noaion X for some process X represens he incremen X k+1 X k wih = { 0,..., M 1 }. As he log-reurns are observable, equaion 16) is he firs measuremen equaion. The second measuremen equaion comes from he observaion of he VIX index level wih error: VIX 2 α VIX 2v + β VIX 2m + γ VIX 2u + δ VIX 2) = ɛ VIX. 20) Jiang and Tian 2007) poin o sysemaic biases in he calculaion of he VIX index, such as model misspecificaion or daa limiaions. For example, in pracice, he index is calculaed using a finie number of opions hereby inducing an error in he compuaion of he inegral presen in he definiion. These biases are capured by he error erm ɛ VIX, which is assumed o follow a normal disribuion wih mean zero and variance s > 0. In order o beer idenify he oal variance of S&P 500 reurns under he P measure, we add a measuremen equaion, which maches he logarihm of he daily Realized Variance RV ) of S&P 500 reurns o he logarihm of he oal spo variance under P, as in Filipović, Gourier, and Mancini 2016). The associaed measuremen error ɛ is condiionally normally disribued wih mean ρ ɛ ɛ 1 and variance c 0 + c 1 RV 1. The raionale behind his componen of he measuremen equaion is 17

19 he following. Andersen, Bollerslev, Diebold, and Ebens 2001), among ohers, provide empirical evidence ha logrv ) is approximaely normally disribued. The condiional mean specificaion of ɛ allows for auocorrelaion in he measuremen error, which can be induced by clusering of price jumps caused by persisence of he price jump inensiy and/or microsrucure noise in he esimaes of daily realized variance. Auocorrelaion in he measuremen error is also repored in Wu 2011). The condiional variance specificaion of ɛ capures in a parsimonious way he heeroscedasiciy of he measuremen error due o he volailiy of realized variance. The daily RV is compued using ick-by-ick daa from he S&P 500 fuures and applying he woscale esimaor of Zhang, Mykland, and Aï-Sahalia 2005) wih fas and slow ime scales given by wo-ick and 20-ick, respecively. The las measuremens are he prices of S&P 500 and VIX opions. We assume ha he opion prices are observed wih an error. This error represens several sources of noise, such as bid ask spreads, iming and processing errors. marke O M,Mk and model prices O M,Mod, M {SPX, VIX}: O SPX,Mod,i Y, v, m, Θ, Θ P,Q ) O SPX,Mk,i O SPX,Mk,i O VIX,Mod,j v, m, Θ, Θ P,Q ) O VIX,Mk,j O VIX,Mk,j We define hese errors as he relaive differences beween = ɛ SPX,opions,i, i = 1,..., N SPX,, 21) = ɛ VIX,opions,j, j = 1,..., N VIX,, 22) where N M, is he number of conracs available in he corresponding marke and he Θ s are he ses of parameers o esimae: Θ P = {κ P v, κ P m, θ P m, ν P m, ν +)P v Θ = {κ v, κ m, θ m, ν v +), ν ), ν v )P, µ +)P Y, µ )P Y, η Y }, v, µ +) Y, µ ) Y }, Θ P,Q = {λ ) 0, λ ) 1, λ ) 2, λ ) 3, λ +) 0, λ +) 1, λ +) 2, σ m, σ v, σ u, κ u, θ u, ρ Y v }. We assume he error erms o be normally disribued and heeroscedasic: ɛ SPX,opions,i N 0, σ 2 ), ɛ VIX,opions ɛ SPX,j,i N 0, σ 2 ), 23) ɛ VIX,j 18

20 The variance of he errors is σ 2 ɛ SPX,i σ 2 ɛ VIX,j = exp φ 0 bid-ask spread i + φ 1 log = exp ψ 0 bid-ask spread j + ψ 1 log K i F SPX T i ) K j F VIX T j ) ) + φ 2 T i ) + φ 3 ), 24) ) + ψ 2 T j ) + ψ 3 ), 25) wih φ i and ψ i in R, i {0,..., 3} Paricle filer A every period = n, he measuremen vecor y collecs observed marke prices. By y = y 0,..., y n ), we denoe all he observaions available up o ime. The filering problem consiss of recursively approximaing he disribuion of he laen sae L, { } L = v, m, u, N +), N ), Z Y +), Z Y ), 26) condiional on y. Paricle filers are perfecly adaped o our problem: They can handle observaions ha are nonlinear funcions of laen variables as well as equaions wih non-gaussian innovaions. There are many ypes of paricle filers. We use he Auxiliary Paricle Filer APF) proposed by Pi and Shephard 1999). Compared o more basic paricle filers, such as he Sampling Imporance Resampling SIR) filer, he APF is beer suied o deec jumps, whereas he SIR filer faces sample impoverishmen leading o poenial paricle degeneracy. Boh filers are described in Johannes, Polson, and Sroud 2009) for filering laen facors from reurns in a Heson model wih jumps in reurns. We develop an exension of heir algorihm ha is able o handle more daa he VIX marke daa on op of he S&P 500 marke daa) as well as he second volailiy facor m, he hird facor for jumps u and he volailiy jumps. The likelihood esimaion and paricle filer are presened in deail in Appendix F. In paricular, we use he weighed likelihood mehod of Hu and Zidek 2002), in order 9 The fac ha he opion pricing errors are normally disribued does no consiue a resricion. The reason is ha he errors are heeroscedasic and he coefficiens generaing he heeroscedasiciy are driven by he daa, i.e., we opimize over he parameers {φ i, ψ i} 0 i 3. 19

21 o assign comparable weighs o S&P 500 and VIX opions. Furhermore, we performed addiional daa reamens for S&P 500 and VIX opions before running he paricle filer. They are described in Appendix G. 5 Esimaion resuls wih and wihou VIX opions This secion analyzes he benefis of including he differen feaures of he full model specificaion, wih and wihou VIX opions in he esimaion daase. We denoe by D 1 he daase conaining S&P 500 opions, S&P 500 reurns and heir daily RV and VIX levels, and by D 2 he daase conaining VIX opions in addiion he componens of D Likelihood crieria and parameer esimaes Before we discuss in deail our parameer esimaes, we would like o sress he fac ha opions are crucial for idenifying he parameers of our model. Even when esimaing highly resriced sub-specificaions of our full specificaion o a daase wihou opions wih he underlying S&P 500 and VIX indices only), we find ha he resuling esimaes of he Q-parameers and some of he P-parameers) have exremely large sandard errors, ypically four o five imes larger han he ones obained wih daases conaining opions. This problem can parly be resolved by exending he in-sample ime period, leading o a more accurae esimaion of he P-parameers, bu no of he Q-parameers. Therefore, he VIX index does no conain sufficien informaion o idenify he Q dynamics of S&P 500 reurns, as has been argued in, e.g., Duan and Yeh 2010, 2011). 10 Table 2 repors he log-likelihood across he model sub-specificaions and he values of he Akaike Informaion Crierion AIC) and Bayes Informaion Crierion BIC), wih boh esimaion daases. [Table 2 abou here.] Inspecion of hese values for D 2 suggess ha he full SVJ3 specificaion is subsanially superior o all nesed sub-specificaions examined. When VIX opions are no par of he esimaion daase 10 Resuls have no been repored for space consrains bu are available upon reques. 20

22 D 1 ), he SVJ3 model only slighly ou-performs he SVJ2 model, where he inensiy of jumps does no load on he addiional facor u. Given ha his facor conrols for simulaneous jumps in he S&P 500 and VIX index, VIX opions conain informaion ha helps idenify he parameers driving is dynamics. This jusifies why he difference in performance beween he SVJ2 and SVJ3 models is larger when VIX opions are included in he esimaion. Oher sub-specificaions resricing m o a consan, or wihou jumps, significanly under-perform he SVJ2 and SVJ3 models for boh D 1 and D 2. A less flexible jump size specificaion, where posiive jumps in he reurns are modeled ogeher wih negaive jumps using a normal disribuion, also under-performs our specificaion significanly. Table 3 presens he poin esimaes and sandard errors resuling from he esimaion of he SVJ3 model o daases D 1 and D 2. [Table 3 abou here.] The esimaed parameers driving he wo variance processes allow idenifying very differen roles. Indeed, v has a high volailiy parameer σ v ranging from 0.57 o 0.76 depending on he model specificaion, wih small sandard error. Besides, i has a high speed of mean reversion under boh measures, implying a half life around 33 days under P and 48 days under Q. In conras, m has a volailiy parameer σ m around 0.10 regardless of he esimaion daase and chosen model, also wih small sandard error. Is speed of mean reversion is difficul o esimae precisely bu ranges beween 0.08 and 0.24, leading o a half life o 3 o 7 years depending on he esimaion daase. We can inerpre he process v as a facor represening erraic shor-erm flucuaions of he variance, whereas he process m is persisen and capures smooher medium- o long-erm rends. 11 The esimaed volailiy of he jump process u, σ u, is close o 0.30, which suggess ha u is no as volaile as v bu also no as persisen as m. Is quie high speed of mean reversion, corresponding 11 Indeed, under he assumpion ha jumps have a minor impac on his expecaion compared o he drif erm, i.e. κ P v >> λ +/ ) k νv P inequaliies saisfied by our parameer esimaes), he condiional expecaion of he variance E P [v T ] can be wrien as: E P [v T ] θm P κ v κ P v + c e κp m T ) m θm) P + e κp v T ) v c m + c κp m θm), P κ P v for a consan c = κv. As κ P κ P v κp v >> κ P m, he coefficiens in fron of v decays much faser han he one in fron of m m. For T equal o hree monhs, e κp v T ) is around 0.16, bu goes down o 0.03 for six monhs, and is of order of magniude of 10 3 for a year. In conras, e κp m T ) is around 0.90 for T equal o six monhs, and as high as 0.80 for one year. Therefore he deviaion of m relaive o is long-erm mean drives he medium- o long-erm expecaion of he variance. 21

23 o a half life beween 60 and 75 days, indicaes ha i capures puncual evens. No surprisingly, we find a prominen leverage coefficien ρ Y v across all models and daases. In line wih he lieraure, he equiy risk premium coefficien η Y is posiive across all models and daases, which implies a posiive diffusive equiy risk premium. The diffusive par of he volailiy risk premium η v = κ v κ P v is always found negaive, which suppors he exisence of a negaive variance risk premium. Jump size esimaes under Q are saisically significan, wih a negaive par which capures he value given o large and rare evens around -11% on average). I is much more negaive han he average negaive jump esimae under P, around 4%. This negaive jump risk premium capures invesors risk aversion o crashes. In conras, posiive jumps have comparable means under boh measures. Similarly o negaive jumps in reurns, posiive jumps in volailiy are priced, wih an average size of around 6% under Q and 2% o 4% under P. Negaive jumps have a zero risk premium by assumpion, bu we find ha heir esimaed size is significanly differen from zero. In fac, resricing i o be equal o zero yields a subsanial loss of likelihood. The inensiy of jumps is mainly driven by he shor-erm volailiy process v, and o a smaller exen by he long-erm process m. 5.2 Filered rajecories In Figure 3, we plo he rajecories of he volailiy processes v and m, filered using he SVJ3 model. Panel A represens he rajecory of v using D 2 as esimaion daase. Unil mid-2007, he volailiy remains a low levels, oscillaing around 10%. When he crisis of he quan-sraegy hedge funds sars in he summer of 2007, he process v gradually goes up o sabilize around 20%. The bankrupcy of Lehman Brohers in Sepember 2008 riggers a sudden increase in v o more han 80%, closely followed by a decrease and a subsequen increase when he House of Represenaives rejecs he Troubled Asse Relief Program a he end of Sepember. The volailiy hen revers back o around 40%, o increase again following he disress of Bank of America in I gradually goes back o a level ha is close o is iniial level, o reach around 10% early The Greek financial 22

24 crisis hen leads o new increases in volailiy, wih a peak around 50% in mid-2010, and a reversion o around 10% following he agreemen on a rescue plan. Noe ha v never ouches or crosses he zero boundary. In Panel B, we plo he difference beween he filered variance rajecories, using D 2 and D 1 as esimaion daases. The esimaed rajecories are consisen hroughou he wo daases in imes of marke calm, bu slighly differ during marke urmoil. The maximum difference is around 10%. These differences can be explained as follows. Firs, he model is likely no o be able o fully represen boh markes during high volailiy imes. Second, he wo markes are subjec o differen fricions, e.g., liquidiy shocks, or consrains and informaion ses of he raders involved, inroducing inconsisencies beween prices. Panel C of Figure 3 represens he filered rajecory of he sochasic cenral endency m resuling from esimaing he SVJ3 model o D 2. The process m is overall more persisen han he process v, in line wih he parameer esimaes found. I sars increasing in mid-2007 from around 1% o around 5%. I sabilizes and oscillaes around ha level for abou a year, unil Sepember 2008 when i increases gradually again, o reach a level close o 10% a he beginning of This increase is followed by a gradual decrease unil beginning of 2010, where m is around 5% again. The high volailiy due o he Greek crisis in 2010 ranslaes ino an sligh increase of m from 5 o 7%. While he process v reaches levels which are close o is iniial level a he beginning of 2010, m sabilizes a a level which is abou 7 imes is iniial level. Panel D displays he difference in he esimaed m rajecory when removing VIX opions from he esimaion daase. The rajecory does no change dramaically, wih a maximum difference of around 4% around he crisis period. [Figure 3 abou here.] In Figure 4, Panel A represens he filered rajecory of he jump inensiy facor u, wih and wihou VIX opions. As he esimaion wih D 2 yields an esimae of λ ) 1 which is much higher han he esimaion wih D 1, he inensiy of jumps as implied by he model becomes oo high and is no suppored anymore by he daa when v is a is peak. The facor u acs as correcion and decreases 23