Estimating the Leverage Parameter of Continuous-time Stochastic Volatility Models Using High Frequency S&P 500 and VIX*

Transcription

1 Esimaing he Leverage Parameer of Coninuous-ime Sochasic Volailiy Models Using High Frequency S&P 500 and VIX* Isao Ishida Cener for he Sudy of Finance and Insurance Osaka Universiy, Japan Michael McAleer Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam The Neherlands and Tinbergen Insiue, The Neherlands and Insiue of Economic Research Kyoo Universiy, Japan and Deparmen of Quaniaive Economics Compluense Universiy of Madrid Kosuke Oya Graduae School of Economics and Cener for he Sudy of Finance and Insurance Osaka Universiy, Japan Revised: May 2011 * The auhors are mos graeful o wo referees for helpful commens and suggesions. The firs auhor wishes o hank Yusho Kaguraoka, Toshiaki Waanabe, and paricipans a he 2010 Annual Meeing of he Nippon Finance Associaion, he CSFI Nakanoshima Workshop 2009, and he Hiroshima Universiy of Economics Financial Economerics Workshop 2010 for valuable commens, and he Japan Sociey for he Promoion of Science (Grans-in-Aid for Scienific Research No ) for financial suppor. The second auhor is mos graeful for he financial suppor of he Ausralian Research Council, Naional Science Council, Taiwan, and he Japan Sociey for he Promoion of Science. The hird auhor is hankful for Grans-in-Aid for Scienific Research No from he Japan Sociey for he Promoion of Science. 1

2 Absrac This paper proposes a new mehod for esimaing coninuous-ime sochasic volailiy (SV) models for he S&P 500 sock index process using inraday high-frequency observaions of boh he S&P 500 index and he Chicago Board of Exchange (CBOE) implied (or expeced) volailiy index (VIX). Inraday high-frequency observaions daa have become readily available for an increasing number of financial asses and heir derivaives in recen years, bu i is well known ha aemps o direcly apply popular coninuous-ime models o shor inraday ime inervals, and esimae he parameers using such daa, can lead o nonsensical esimaes due o severe inraday seasonaliy. A primary purpose of he paper is o provide a framework for using inraday high frequency daa of boh he index esimae, in paricular, for improving he esimaion accuracy of he leverage parameer,, ha is, he correlaion beween he wo Brownian moions driving he diffusive componens of he price process and is spo variance process, respecively. As a special case, we focus on Heson s (1993) square-roo SV model, and propose he realized leverage esimaor for, noing ha, under his model wihou measuremen errors, he realized leverage, or he realized covariaion of he price and VIX processes divided by he produc of he realized volailiies of he wo processes, is in-fill consisen for. Finie sample simulaion resuls show ha he proposed esimaor delivers more accurae esimaes of he leverage parameer han do exising mehods. Keywords: Coninuous ime, high frequency daa, sochasic volailiy, S&P 500, implied volailiy, VIX. JEL Classificaions: G13, G17, G32. 2

3 1. Inroducion The negaive correlaion beween reurn and is volailiy is one of he mos salien empirical feaures of ime series of equiy price observaions. Many varians of he coninuous-ime and discree-ime sochasic volailiy (SV) and GARCH-ype volailiy models incorporaing his feaure in he dynamic equaion for volailiy have been proposed in he lieraure. This correlaion in he underlying asse price or index affecs he heoreical prices of opions in such a way as o fi and explain parially he empirically observed skew paerns in he Black-Scholes opions implied volailiies ploed agains he srike prices 1. Thus, such a correlaion has araced subsanial aenion in he asse pricing and financial economerics lieraure. Saisical esimaion of his correlaion for a paricular ype of coninuous-ime SV models is a primary focus of his paper. The negaive price-volailiy correlaion is cusomarily referred o as leverage afer Black s (1976) explanaion based on he increased deb-equiy raio of a firm following is share price decrease raising is share price volailiy 2. In his paper, we also use he erm leverage inerchangeably wih correlaion beween reurn and is volailiy, wihou resricion regarding is sign. For he analysis of derivaives, one-facor mean-revering diffusion processes ofen augmened by jump componens are commonly used as coninuous-ime models of he spo variance, among which he affine-drif square-roo SV model of Heson (1993) enjoys populariy due o is analyical racabiliy. One-facor SV diffusion models incorporae leverage by allowing he wo Brownian moions driving he price process and is volailiy process, respecively, o be correlaed. Even if he chosen parameric model is correcly specified, i requires an accurae esimae of his correlaion,, or he leverage parameer, ogeher wih he oher parameers, for he model o be useful in derivaives pricing and hedging. 1 See, for example, Das and Sundram (1999). 2 For economic explanaions in he equiy case, see Bollerslev e al. (2006) and he references cied herein. The leverage concep does no apply o non-equiy cases. 3

4 In his paper, we propose a new mehod for esimaing his leverage parameer, eiher individually or joinly wih he oher parameers, for a class of coninuous-ime SV models using high frequency inraday observaions of he price and is model-free opions implied volailiy. Essenially, we propose o use he realized leverage, or he realized correlaion beween he price and he model-free implied variance, for improving saisical inference 3. The realized correlaion beween wo series is he realized covariaion divided by he produc of he wo realized volailiies. Alhough many papers, including Barndorff-Nielsen and Shephard (2004) and Vorelinos (2010), have sudied he realized covariaion and correlaion beween a pair of asse reurn series, he use of hese quaniies o measure leverage and o esimae he parameers of specific SV models, o he bes of our knowledge, is novel. We focus on he Heson SV model in his paper for wo reasons. Firs, i is one of he mos commonly used and imporan models among SV models. Hence, i is more han an example for merely illusraing he benefi of using high-frequency implied volailiy daa. Second, when i is correcly specified and here are no measuremen errors, he realized leverage converges o in probabiliy as he ime inervals beween observaions shrink o zero, even if he lengh of he whole sample period is fixed. This propery, ogeher wih he availabiliy of analyical expressions for he condiional momens of realized variance, makes he Heson SV model a simple and clear example for showing he benefis of using he high frequency implied volailiy daa joinly wih he S&P 500 index daa. Alhough canno be backed ou in his way for models oher han he Heson SV model, using high frequency observaions of boh indices is likely o produce superior parameer esimaes. Under he affine-drif consan-elasiciy-of-variance (CEV) SV process, he realized leverage is no consisen excep under he Heson SV special case, bu appears o be robus as an esimaor of. For more exac inference for non-heson models, simulaed momens of realized measures, including he realized covariaion or correlaion, may be used. 3 For esimaing models oher han he Heson SV model, i may be more convenien o use he momens of he realized covariaion raher han he realized correlaion. 4

5 Inraday high frequency daa have become readily available for an increasing number of financial asses and heir derivaives in recen years. However, i is well known ha aemps o apply direcly popular coninuous-ime models ha are inended o approximae financial processes a daily or lower frequencies o shor inraday inervals of one hour or less, and esimae he parameers using high-frequency reurns, say five-minue reurns, lead o nonsensical parameer esimaes due o inraday seasonaliy 4 and various microsrucure effecs. For direcly modeling reurns a shor ime inervals, simple jump diffusion models are clearly no an accurae approximaion, so ha i is necessary o use fundamenally differen approaches, such as he one pursued by Rydberg and Shephard (2003). However, such approaches, while imporan in empirically undersanding microsrucure phenomena, do no easily lend hemselves o he analysis of derivaives. While mos of he ime series esimaors for simple coninuous-ime SV models proposed in he lieraure are implemened using daily daa, due no only o he limiaion of daa availabiliy bu also o he above consideraion, some auhors have sough o exrac informaion conained in high frequency inraday daa for parameer esimaion and jump idenificaion, reaining simple jump diffusion models. Noable among hem is Bollerslev and Zhou (2002), who proposed a GMM esimaor for he Heson model and is several exensions using momen condiions based on condiional momens of he daily realized variance, which is a daily aggregae of shor inraday squared reurns. For esimaing he BZ esimaor relies on he cross momen of he daily closing price and he daily realized variance. The resuls of our finie sample simulaion experimens using he Heson model indicae ha heir GMM esimaor for is severely biased oward zero. BZ s approach of coninuous-ime model esimaion based on momens of daily realized measures has been followed in several sudies, and he presen paper belongs o his srand of he lieraure. Corradi and Disaso (2006), hereafer referred o as CD, proposed o use uncondiional momens and auocovariances of he realized variance 4 See, for example, Andersen and Bollerslev (1997), who documened empirical evidence of pervasive inraday periodiciy in asse reurn volailiy. 5

6 and relaed realized measures for a similar GMM esimaion procedure, bu hey did no consider he esimaion of. BZ and CD used only he high frequency observaions of he price process. Garcia e al. (2011), hereafer referred o as GLPR, proposed o use wo ses of momen condiions, including higher-order ones, for GMM esimaion of he Heson model: one se involving he daily asse reurns and daily realized measures calculaed using high-frequency inraday asse reurns, and anoher se involving he daily asse reurns and he daily observaions of he model-dependen implied volailiy. As wih his paper, one focus was on he esimaion of. The finie sample simulaion resuls provided in GLPR and in his paper indicae ha he GLPR esimaor provides a vas improvemen over he BZ esimaor in erms of accuracy in esimaing of he Heson SV model wih nonzero leverage. We demonsrae by simulaion ha he use of high frequency inraday implied volailiy index daa would lead o furher subsanial gains in efficiency. The CBOE s S&P 500 implied (or expeced) volailiy index (VIX) is designed o measure he volailiy of he S&P 500 index wihou relying on a paricular opion pricing model, such as he Black-Scholes or Heson models. Many auhors have aemped o exploi informaion in VIX in esimaing models for he S&P 500 index. Under he assumpion ha he S&P 500 index follows an affine-drif SV process (possibly wih cerain ypes of jumps), VIX is an affine ransformaion of is spo variance. We are no he firs o ake advanage of his relaion. Based on his relaion, Duan and Yeh (2010) proposed an esimaor for he affine-drif CEV SV model wih Poisson-ype price jumps for he S&P 500 index, using daily observaions of boh he S&P 500 index and VIX. However, hey only used daily daa o esimae a discreized version of he model. Aï-Sahalia and Kimmel (2007) also used he VIX joinly wih he S&P 500 daa in implemening heir approximae maximum likelihood esimaor. Bakshi e al. (2006) and Dosis e al. (2007), among ohers, ake he VIX process as he objec of direc ineres raher han reaing i as an insrumen o esimae he underlying volailiy process, and used daily VIX observaions o esimae he coninuous-ime SV models for VIX. However, none of hese sudies used daa of frequencies higher han a day. 6

7 Applicabiliy of he proposed approach is no limied o he S&P 500 index. If here exiss a liquid opions marke for he underlying process of our ineres, wih a wide specrum of srike prices, and he inraday high frequency daa of heir prices were available, we may calculae he model-free implied volailiy values a a high enough frequency for he applicaion of our proposed approach. For many financial series, he implied volailiy calculaion sep is convenienly done by exchanges and oher insiuions. On he heels of he success of VIX, he universe of model-free implied volailiy indices, as well as exchange-raded opions and fuures on hese volailiy indices, has been expanding rapidly in recen years. The CBOE now calculaes and disseminaes volailiy-relaed indices for a variey of financial marke indices, and currency and commodiies ETFs, including he CBOE NASDAQ-100 Volailiy Index, CBOE EuroCurrency Volailiy Index, CBOE Crude Oil Volailiy Index, CBOE Gold ETF Volailiy Index. The CBOE and he Chicago Mercanile Exchange (CME) Group work ogeher o provide he CBOE/NYMEX Crude Oil (WTI) Volailiy Index and CBOE/COMEX Gold Volailiy Index, applying he CBOE VIX mehodology o he prices of opions on crude oil and gold fuures. They also inend o provide he CBOE/CBOT Soybean Volailiy Index and Corn Volailiy Index. The Deusche Börse provides he VDAX-NEW index for he DAX, and Osaka Universiy, Japan, provides he VXJ and CSFI-VXJ for he Nikkei 225 index (see Fukasawa e al. (2010) for he laer indices). Various insiuions calculae and updae model-free implied volailiy indices for oher indices, alhough he updaing frequency is no always high enough for our purpose. Anoher conribuion of his paper is a proper adjusmen of he momen condiions o reflec he fac ha daily realized measures are calculaed only for he rading hours ha do no cover a full day. In esimaing he Heson model for share prices of individual socks or he S&P 500 index, CD and GLPR rea he six and a half hours (9:30 am - 4:00 pm) for which NYSE is open as a full day as if overnigh hours were non-exisen. Their closed-form momen condiions for he Heson SV case clearly need o be modified, considering he overnigh marke closure (nearly hree quarers of a day). 7

8 Oherwise, he esimaor will be biased. We corroborae his claim by firs driving he modified momen condiions allowing for overnigh marke closure, and hen performing Mone Carlo simulaion using he BZ momen condiions. The plan of he remainder of he paper is as follows. Secion 2 develops a framework for using inraday high frequency implied volailiy indexes daa for SV model esimaion, in paricular, a leverage esimaor using realized measures of price and volailiy indexes for he Heson SV special case. Secion 3 presens some finie sample simulaion resuls. Secion 4 analyzes he empirical resuls using inraday high frequency S&P 500 and VIX. Secion 5 gives some concluding remarks. 2. Esimaion of leverage and oher parameers using realized measures of boh he price and implied volailiy index Consider he following class of affine-drif SV diffusion processes: dp V db (1) db dw 1 dw (2) V 1 dv V d dw (3) where p is he log price process, W W are Brownian moions independen of 1 2 each oher, and V is called he spo variance process 5. The parameers and deermine, respecively, he speed of variance mean reversion and he average level of he spo variance. As db dw 1 d is he so-called leverage parameer. When he diffusion coefficien V of he variance process (3) is of he form V wih 0, i is called he affine-drif CEV diffusion. The affine-drif CEV wih 05 is Heson s (1993) square-roo SV model, and he affine-drif CEV wih 5 The drif funcion of he price equaion (1), which is irrelevan for our analysis, is se o be zero, as in BZ. 8

9 10 is Nelson s (1990) GARCH SV diffusion. A key elemen in consrucing he proposed esimaor is he well-known fac ha, for he above SV model, he following relaion holds beween he risk-neural expecaion of he inegraed variance over any horizon, 0 Q v E V sds V (4) a each poin in ime, where 0 and are consans ha depend on and he parameers of he model, boh under he physical and risk-neural measures (see, for Q example, Duan and Yeh (2010)). neural measure, condiional on he filraion on E is he expecaion operaor under he risk F, and F is he filraion on he probabiliy space on which p is defined 6. Noe ha i is he affine form of he drif funcion, raher han a paricular form of he diffusion funcion (such as he square-roo diffusion funcion in he Heson SV model), of he spo variance process ha gives rise o he affine relaion, namely equaion (4). The VIX index, a widely wached sock marke volailiy indicaor ha was inroduced by he Chicago Board Opions Exchange (CBOE), is inended o approximae v a 30 of he S&P 500 index process, using he heoreical formula in he model-free implied volailiy lieraure (see Brien-Jones and Neuberger (2000), Demeerfi e al. (1999), Jian and Tian (2005)), linking he marke prices of a cross-secion of opions on he S&P 500 index and v (see CBOE (2009)). In he discussion below, we fix 30 wrie v for v 30, and rea VIX 2 v as an exac relaionship 7, which makes he spo variance observable up o an affine ransformaion 6 An affine relaion, albei wih differen values of he wo consans, beween he expecaion of he inegraed variance and he spo variance, holds under he physical measure as well, which is widely used in deriving analyical expressions for some of he momen condiions by BZ, GLPR, and his paper. 7 Aï-Sahalia and Kimmel (2007) simply used VIX o approximae he lef-hand side of equaion (4) 9

10 wih unknown parameers and. For he S&P 500 index, he CBOE calculaes and disseminaes he VIX index on a real-ime and inraday very high frequency basis, so ha we do no have o collec S&P 500 index opions ick daa for he calculaion of v. If here is a liquid marke for opions wrien on he process of ineres, wih a reasonably wide and dense cross-secion of srikes, a VIX-ype model-free implied volailiy may be calculaed for financial insrumens oher han he S&P 500 index. If high frequency observaions of he price process and a VIX-ype index, or opion prices necessary o calculae such an index, were available, he realized leverage could be calculaed. Hence, he discussion below also applies o financial processes in addiion o he S&P 500 index. In he empirical secion, we use inraday VIX daa. Define q T q V T Vs ds (5) and T T 1 V V (6) Under he SV model, equaions (1)-(3), he realized variance T is such ha: RVT for he ime inerval T N 2 it N i 1T N V T (7) RV p p i1 p where p denoes convergence in probabiliy as he number of observaions, N, during he fixed ime inerval T goes o infiniy. wihou invoking he heory of he model-free implied variance. 10

11 For esimaing he parameers of he SV model, BZ and CD suggesed using momens of he daily realized variance 8 log index level, (BZ also suggesed adding he cross momen of he daily p, and he daily realized variance for esimaing joinly wih he oher parameers of he Heson model), and GLPR recommended using higher-order momens of he daily realized variance or daily (model-dependen) implied volailiy. For esimaing joinly, GLPR proposed adding he (higher-order) cross momens of he daily log reurn p p 1 (raher han he log level p ) and he realized variance (or he implied volailiy), which is noably differen from BZ s choice of he cross momen. Essenially, our new proposal uses various momens of he realized measures of he model-free implied volailiy index and he realized covariaion/correlaion beween he underlying index and he model-free implied volailiy index, defined below, for conducing saisical inference on he SV process 9. In order o subsaniae our claim abou he benefi of using high frequency implied volailiy daa, we will focus on he Heson SV example. For SV models oher han he Heson SV model, we can use he simulaed mehods of momen (SMM) esimaor wih momens of he realized measures calculaed by simulaion since analyical expressions for he momens are, in general, unavailable 10. We have for he realized variance, RVVT, of v, and he realized covariaion, RCOVT, beween p and v : N 2 p T Vs (8) RVV v v ds T i T N i T N i1 8 CD also suggesed versions wih oher realized measures, such as he realized bipower variaion replacing he realized variance. 9 One could also use realized power variaions oher han he realized variance. 10 I may no be compuaionally feasible o calculae he condiional momens a each poin in ime for each ieraion of he objecive funcion minimizaion. The SMM esimaor of CD based on uncondiional momens (including auocovariances) may be used insead. 11

12 N T 1/2 1 1 s s (9) RCOV v v p p V V ds T i T N i T N i T N i T N i1 p We can also define he realized correlaion, or he realized leverage, as follows: RCORR T T RCOV p T * RVV T RV T V T T s V V 2 1/2 s V s ds ds (10) Noe ha is cancelled ou. If we assume ha he volailiy of he spo variance is of he CEV funcion, V V 2 2, we obain RVV 2 T p V, p 1 * 1 RCOV T V T, and V T 2 / V T 2 V T. Noe ha 2 is cancelled ou furher. For he special case of he Heson SV process (he affine drif CEV SV wih 05 ), hereby leading o a key resul 11 : * T p RCORRT. (11) This means ha, in he coninuous-record limi wih a fixed T, he leverage parameer can be recovered wihou saisical uncerainy if he Heson SV specificaion is correc. The consisency resul (11) may no hold in realiy due o a variey of facors, such as microsrucure noise, and he relaion, RCORR T is no exac, hence is no a deerminisic consrain, for finie N and T, even in he absence of microsrucure noise. Neverheless, RCORR T may perform well as an esimaor of he leverage parameer,, under he Heson SV process. The inerval over which he quaniies are measured a high frequency is defined o be T in he above, for noaional simpliciy. However, (11) clearly holds when he measuremen period is a collecion of 11 For he Heson SV case, he cenral limi heorem also obains (as N wih T fixed) as a sraighforward applicaion of Theorem 2 of Barndorff-Nielsen e al. (2006): N RCORR V T 2 / V, T, T Law N 2 0, 1 12

13 subinervals K1 K, where 1 K1 K T if he hree realized measures ha comprise he realized correlaion are defined over he same se of subinervals and he observaion inervals shrink o zero in each subinerval. This is convenien as mos financial markes have inerrupions in rading, such as overnigh hours, holidays, and weekends. For non-heson cases, we recommend using momens of he realized covariance or correlaion, and esimae, joinly wih he oher parameers by GMM or SMM, as he sochasic quaniy * o which RCORR T converges in probabiliy, is no equal o, and * T T 2 s s V T Vs 1/2 holds as V V ds ds, by he Cauchy-Schwarz inequaliy. However, we repor simulaion resuls in he nex secion where, for he affine-drif CEV case, RCORR T, even when 1 or 15 We also have: p T 2 2 T T Vs ds T RVV RV V (12) for fixed T 2 2 he righ-hand side of which becomes V 2 affine-drif CEV SV and T V under he 2 2 (a consan) under he Heson SV model. esimaing he Heson SV model, his high frequency (wih a fixed-t) asympoic relaion should be paricularly helpful if is o be esimaed joinly. This is so because, as is he case wih esimaing by he realized leverage, for he Heson SV model can be recovered wihou saisical imprecision in he coninuous-record limi for fixed T, under ideal condiions. T For Furhermore, we have he following resuls involving and : p v, T 1 V T T, (13) 13

14 p 2 2, T 2 T 2 2 T v V V T (14) where N T q v q v (15) T, i T N N i1 These resuls may be exploied in join esimaion for. The addiional parameers, and may be informaive abou he parameers of he SV process under he risk-neural measure, and hence also he volailiy risk premium, bu are nuisance parameers if he ineres is only in esimaing he parameers of he SV model under he physical measure. In his paper, we do no pursue he use of hese relaions furher. In he remainder of he paper, we focus on he Heson SV example, and consider exending he BZ esimaor 12. BZ showed ha, for he special case of he Heson SV model where he variance diffusion is given by 13 : 1 dv V d V dw (16) he following analyical expressions for he condiional momens of V 1 hold: E V E V (17) E V E V I E V J (18) V 1 2 b1 V 1 b1 E p 1 1E p 1 1 p a 1 a 1 (19) 12 Alernaively, one can ake he GLPR esimaor wih higher-order momens as a saring poin, bu making non-full-day adjusmens o he analyical expressions for he higher-order condiional momens would be more involved. 13 Noe ha hese equaions reflec he correcions by Bollerslev and Zhou (2004) for he original equaions in BZ. 14

15 where V a1 E p b a e 1 a 1 1 e b a A 1 e 2e B e e e C e e D 1 e I a C b a A (20) J I b a D 2a b A 1 b B (21) and E : E G is he expecaion operaor under he physical measure, 1 condiional on : s1, s1 ; s, 1, G V, he discree filraion, or he sigma algebra generaed by he daily realized variance series. The noaion G is inroduced here in preparaion for he case of non-full-day rading sessions. For esimaing of he Heson SV, BZ proposed a GMM esimaor (GMM-BZ1), using he sample analogues of he following se of momen condiions: EE V V (22) 2 2 EE V V (23) E E V V V (24) E E V V V (25) E E V V V (26) E E V V V (27) 15

16 V 1 2 b1 V 1 2 b 1 E E p1 p 1 0 a 1 a1 (28) where he uncondiional expecaion E is aken under he physical measure. BZ suggesed simulaions for calculaing he condiional momens if he model being esimaed is a non-heson SV and suiable closed-form expressions are no available. As repored in he nex secion, he resuls of our Mone Carlo simulaion experimens indicae ha, when esimaed joinly by GMM-BZ1 wih he oher parameers of he Heson SV, is severely biased. For esimaing of he Heson SV model, we propose wo new mehods, namely: (i) esimae all he parameers and joinly by GMM using he sample analogues of equaions (22) - (27) and he realized leverage formula: RCORR T (29) replacing he sample analogue of (28) (called GMM-BZ-RL); or (ii) esimae by ˆ RCORR (called ˆ RL) and by BZ s original GMM esimaor, using T he sample analogues of equaions (22) - (27) (called GMM-BZ2). I may be possible o derive he condiions for he SV process, he measuremen error process, and he relaive rae of N and T, under which he esimaors, GMM-BZ-RL and RL, are consisen, along he lines of CD. The realized bipower variaion counerpars jump-robus esimaors for RBVT and RBVVT o RVT and RVVT, respecively, are T 2 V and V T s ds even if he price (1) and he spo variance process (3) conain cerain ypes of jumps. CD proposed o use RBVT for a jump-robus specificaion es of he diffusion componens of a jump-diffusion model. RCOVT may be affeced, even asympoically ( N ), by jumps if he price jumps and volailiy jumps arrive simulaneously (see Jacod and Todorov (2010) for empirical 16

17 evidence of price-volailiy cojumps in he S&P 500 index). I is also possible o consruc a realized measure ha serves as a jump-robus esimaor for T V 1/2 s V s ds. We may also use a Lee-Mykland-ype esimaor (Lee and Mykland (2008)) o esimae direcly and remove jumps from he observaions. We leave hese as opics for fuure research 14. A major complicaion in esimaing a model of a financial process is ha high frequency inraday observaions used for consrucing realized measures ofen do no cover an enire rading day. For example, he S&P 500 cash index value is observed only for he period 9:30-16:00 per rading day, which is less han one-hird of a day. In applying GMM esimaors wih ses of momen condiions involving realized measures o individual sock prices and he S&P 500 index, CD and GLPR ignore he exisence of overnigh non-rading hours. Treaing 6.5-hour daily realized measures as if hey were 24-hour flow quaniies, and ignoring he evoluion of he price and is sochasic volailiy processes during overnigh hours, lead o incorrec analyical expressions for he momens as funcions of he unknown parameers and observables. Hence, we modify equaions (22) - (27) as follows, aking he marke closure (16:00-9:30) ino consideraion: EE V V (30) 2 2 EE V V (31) E E V V V (32) E E V V V (33) E E V V V (34) E E V V V (35) 14 We would need o carefully consider how he spo/implied volailiy relaion, equaion (4), is affeced by various ypes of jumps. 17

18 where E : E G. For example, observaions of he S&P 500 index for a rading day are from ime period (9:30) o (16:00) ( 0 27 ). Condiioning on he informaion available a he session s opening, raher han he session s closing, makes he derivaion of condiional momens and he resuling expressions much simpler. Following he derivaion of equaions (17) and (18) by BZ for he case of 1 (24-hour rading), i is sraighforward o obain: E V E V (36) E V E V I E V J (37) where 1 I a C b a A J I b a D 2a b A 1 b B See he appendix for a derivaion of he above relaionships. Noe ha, alhough equaions (36) and (37) appear o be virually idenical o BZ s 24-hour rading versions, namely equaions (17) and (18), hey are differen. The I and J given above are modificaions of BZ s I and J, allowing for non-full-day rading sessions over which he inegraed variance is defined, and he resuling ime gap beween (he end of he period over which V is defined) and 1 (he beginning of he period over which V 11 is defined). If 1 equaions (36) and (37) reduce o (17) and (18), respecively. We may use he sample analogues of equaions (30) - (35) and T 1 RCOV RVV RV in consrucing momen condiions for GMM esimaion. Our adjusmen mehod 18

19 assumes ha he spo variance follows he same Heson SV during rading hours and overnigh hours. Admiedly, his is an unrealisic assumpion. We could consider lowering he average variance for nigh hours, bu leave i as a fuure research opic. Delving oo deeply ino seasonaliy issues would defea he purpose of using daily aggregae quaniies for esimaing simple coninuous-ime SV models ha have proved useful as approximaions of financial processes a he daily or weekly measuremen inervals. Finally, noe ha he realized leverage wihou any adjusmen for 1 converges o in probabiliy, even if he spo variance process follows he Heson SV wih differen ses of values during rading hours and during overnigh non-rading hours, if remains he same. 3. Finie sample simulaion resuls In his secion, we repor he resuls of Mone Carlo simulaion experimens o examine he finie sample properies of he BZ GMM esimaor and he proposed esimaors for he Heson model for he case 0 Noe ha, alhough BZ derived momen condiions for he Heson SV model wih 0, and exended he resuls o he Heson SV model wih price jumps and wo-facor SV models, hey only conduced heir experimens for he case of he Heson SV model wih zero leverage, 0 The sample pahs of p and V are simulaed by he Euler-Maruyama scheme ( or 30 seconds) 10,000 imes. The lengh, T, in days of each simulaed pah is 960 as in GLPR, afer he observaions from a burn-in period of 240 days are discarded. A he sar of he burn-in period, V is se o he long-run average of he spo variance. As in BZ and GLPR, he uni ime is a day raher han a year. Noe ha we do no observe ransformaion, V, in pracice. However, as we rea is affine v, as observable under he affine-drif SV process and he exra parameers, and, of he ransformaion are cancelled ou in he realized leverage ha is calculaed using observaions of p and v, he simulaion resuls would be he 19

20 same if we were o use observaions of v V regardless of he values assigned for and Hence, in order o simplify he experimens, we choose 1 and 0 hereby making V observable. (i) Daa from coniguous full-day rading sessions We firs examine he scheme in which daily rading sessions las 24 hours and here are no breaks beween daily sessions. The values of p and v V are observed once every five minues, and daily realized measures are calculaed once a day, using 288 five-minue log price reurns and differences in V. GLPR used a simulaion scheme ha is similar o ours, bu hey divided each day ino 80 5-minue observaion inervals, which are effecively 18-minue inervals. The firs se of rue parameer values is (1, 25, 1, 5) which corresponds o Parameer Se A in GLPR. The long run spo variance, 25 which is he value se in BZ, is abou 7.75% per annum if one year has 240 days. The second se of rue parameer values is idenical o he firs, excep ha 05 which induces a slower mean reversion of he spo variance. The parameers are esimaed using observaions from each of he simulaed sample pahs of p or p V. For all of our GMM esimaors, we use he opimal covariance marix esimaed by he Newey-Wes scheme wih five lags, as in BZ, and impose he saionariy condiion, 2 2. The resuls for he firs se of rue parameers are summarized in Panel A of Table 1. Noe ha he biases and RMSEs shown are muliplied by 100. Boh he bias and he RMSE of ˆ and, o a lesser bu sill serious degree, he RMSE of ˆ of GMM-BZ1, are so large as o render GMM-BZ1 meaningless and undesirable. The performance of he proposed esimaor GMM-BZ1-RL relaive o GMM-BZ1 is beer overall in erms of biases and RMSEs, excep for a slighly larger bias in esimaing and is vasly 20

21 superior as an esimaor of. In fac, he bias and RMSE of ˆ are only.0007 and.0016, respecively, which are negligible compared wih hose produced by GMM-BZ1. More imporanly, he performance of GMM-BZ1-RL in esimaing appears o be subsanially beer han he GLPR esimaors 15. As does no ener equaions (22) - (27), and he oher parameers do no ener (29), here may be no efficiency gains in esimaing joinly wih he oher parameers by GMM raher han separaely by using he realized leverage. This, in fac, appears o be he case, as is corroboraed by RL s even smaller bias and RMSE as an esimaor of The performance of GMM-BZ2 as an esimaor of is comparable o ha of GMM-BZ-RL. The overall paern in he resuls of he experimen using he second se of rue parameer values is similar o he previous case. (ii) Daa from non-full-day rading sessions In his subsecion, we invesigae he effecs on he GMM esimaor of no properly correcing he momen condiions for he exisence of marke closure beween rading sessions. We assume ha he log price and he variance follow he Heson SV process, wih.1,.25,.1,.5, boh day and nigh, and are observed only for he firs six hours of each day. We assume ha he economerician reas he observed six hours of daa as arising from he firs h hours of each day ( h or 24 ), and esimaes ses h 24 by GMM, wih he sample analogues of he momen condiions in equaions (30) - (35), and by he realized leverage. I is noed ha h 6 is correc, and h 24 ignores 18 hours in beween sessions, in addiion o incorrecly reaing 6 hours as 24 hours. required in compuing he realized leverage for he cases wih 1. Noe ha no adjusmen is 15 Noe, however, ha a day is divided ino 288 subinervals in our scheme, while here are only 80 in heirs. On a separae issue, a comparison of our simulaion resuls for GMM-BZ1 and GLPR s resuls for heir esimaors reveals ha he efficiency gains of he GLPR esimaors, relaive o GMM-BZ1, likely arise no only because of he use of implied volailiies, bu also hrough he use of higher-order momens and/or he use of asse reurns (raher han asse prices) in consrucing cross momens. 21

22 The resuls are shown in Table 2. The biases and RMSEs increase as h deviaes from h 6, excep for. Our simulaion resuls indicae ha here is a serious need for adjusmens. (iii) Realized leverage under he affine-drif CEV wih 05 In his subsecion, we repor he resuls of Mone Carlo simulaion experimens for he realized leverage as an esimaor of, when he affine-drif CEV SV, equaions (1) - (3) wih V 5 generaes he daa p v 16. This exercise is o check he robusness of he realized leverage esimaor for under he misspecificaion of Heson SV model s square-roo diffusion funcion. The resuls should no be consrued as our recommendaion for he realized leverage as a direc esimaor for non-heson cases. Recall ha he realized leverage under he affine-drif CEV 1 converges in probabiliy o V V V he absolue value of which is T 2 T 2 T smaller han he rue, unless 5 The seup is idenical o he firs seup ( 1) used for he coniguous full-day rading sessions case, excep ha he CEV exponen,, is se o be 10 (GARCH SV) and 15 The resuls for 10 and 1.5, ogeher wih hose for he Heson SV case of 5 invesigaed above, are summarized in Table 3. The biases and RMSEs for he wo non-heson cases are larger han he very small values for he Heson SV, bu are 1 neverheless sill small. This implies ha he parameric configuraions ha have been chosen here. V T 2 V T 2 V T 1, a leas under 4. Empirical resuls for inra-day high frequency S&P 500 and VIX 16 Jones (2003), Aï-Sahalia and Kimmel (2007), Duan and Yeh (2010), and Engle and Ishida (2002), among ohers, have repored empirical evidence poining o 0.5 for equiy reurns daa using he CEV SV and relaed models. 22

23 As an empirical illusraion 17, we nex consider applying he proposed esimaors o he Heson SV model using inra-day ick daa for he S&P 500 index and VIX. The daa for boh series are obained from TickDaa, and he sample period is from Sepember 22, 2003 hrough o December 31, 2007 (giving 1,077 rading days). Based on a visual inspecion of he volailiy signaure plos of he S&P 500 and he VIX daa in Figures 1 and 2, we choose five-minue inervals o calculae inra-day log differences of he S&P 500 series and he differences in he VIX squared series o alleviae he effecs of microsrucure noise. The raw VIX daa in annualized percenages are scaled o daily percenages and are squared before five-minue incremens are aken. The realized leverage obained is The resuls of he join GMM esimaion of by GMM-BZ2, and by GMM-BZ-RL, are summarized in Table 4. The sandard errors are he usual asympoic GMM sandard errors. We need o be careful in inerpreing he sandard errors given o he esimaes by GMM-BZ-RL as we have no ye esablished he asympoics for his esimaor. I is likely ha, in a double asympoic framework, T and 0, he componen of he esimaor GMM-BZ-RL is consisen for a a faser rae in he absence of measuremen errors. When we rea he daa as arising from coniguous 24-hour sessions, h 24 he parameer of he long-run variance, is esimaed o be.3647 by GMM-BZ-RL and.3624 by GMM-BZ2, which is no very differen from he average RV, When we rea he daa as arising from non-coniguous 6.5-hour rading sessions, is esimaes are much larger (namely, by GMM-BZ-RL and by GMM-BZ2). The volailiy-of-variance parameer, is also esimaed o be much larger under he correc h 65 assumpion han under he incorrec h 24 assumpion. These 17 As a model for he processes of equiy indices, he Heson SV specificaion has repeaedly been rejeced in favour of more complicaed models; see, for example, Andersen e al. (2002), Aï-Sahalia and Kimmel (2007), Jones (2003), Chernov e al. (2003), Pan (2002), Eraker e al. (2003), and Eraker (2004). Hence, we mus go beyond esimaing he Heson SV model for conducing a full-fledged empirical analysis. 23

24 differences would ranslae o large differences in heoreical opion prices. 5. Conclusion We have proposed o use inraday high frequency model-free implied volailiy daa in consrucing realized-measures-based momen condiions, in paricular, cross-momen condiions, of he GMM/SMM esimaor for coninuous-ime SV models of asse price processes. We have focused aenion on Heson s affine-drif square-roo SV model, and proposed he realized leverage as an esimaor for he leverage parameer, which is shown by simulaion experimens o deliver accurae esimaes of he leverage parameer under his model. We also demonsraed by simulaion experimens he imporance of making proper adjusmens o he momen condiions when realized measures are compued using daa from non-coniguous non-full-day rading sessions. Analyical expressions for momen condiions are usually no available oher han for he Heson SV specificaion, bu he simulaed mehod of momens (SMM) approach may be used. Our argumen for he use of inraday implied volailiy applies no only o he S&P 500 index, bu also o hose equiy price indices for which inraday high-frequency observaions of a VIX-like model-free implied volailiy are available. 24

25 References [1] Aï-Sahalia, Y., and R. Kimmel, 2007, Maximum likelihood esimaion of sochasic volailiy models, Journal of Financial Economics 83, [2] Andersen, T.G., L. Benzoni, and J. Lund, 2002, Esimaing jump-diffusions for equiy reurns, Journal of Finance 57, [3] Andersen, T.G., and T. Bollerslev, 1997, Inraday periodiciy and volailiy persisence in financial markes, Journal of Empirical Finance 4, [4] Bakshi, G., N. Ju, and H. Ou-Yang, 2006, Esimaion of coninuous-ime models wih an applicaion o equiy volailiy dynamics, Journal of Financial Economics 82, [5] Barndorff-Nielsen, O.E., S.E. Graversen, J. Jacod, and N. Shephard, 2006, Limi heorems for bipower variaion in financial economerics, Economeric Theory 22, [6] Barndorff-Nielsen, O.E., and N. Shephard, 2004, Economeric analysis of realized covariaion: High frequency based covariance, regression, and correlaion in financial economics, Economerica 72, [7] Black, F., 1976, Sudies in sock price volailiy changes, Proceedings of he 1976 Meeings of he American Saisical Associaion, Business and Economics Saisics, [8] Bollerslev, T., M. Gibson, and H. Zhou, 2011, Dynamic esimaion of volailiy risk premia and invesor risk aversion from opion-implied and realized volailiies, Journal of Economerics 160, [9] Bollerslev, T., J. Livinova, and G. Tauchen, 2006, Leverage and volailiy feedback effecs in high-frequency daa, Journal of Financial Economerics 4, [10] Bollerslev, T., and H. Zhou, 2002, Esimaing sochasic volailiy diffusion using condiional momens of inegraed volailiy, Journal of Economerics 109, (2004, Corrigendum, Journal of Economerics 119, ). [11] Brien-Jones, M., and A. Neuberger, 2000, Opion prices, implied price processes, and sochasic volailiy, Journal of Finance 55, [12] CBOE, 2009, The CBOE volailiy index - VIX, CBOE websie. 25

26 [13] Chernov, M., A.R. Gallan, E. Ghysels, and G. Tauchen, 2003, Alernaive models for sock price dynamics, Journal of Economerics 116, [14] Corradi, V., and W. Disaso, 2006, Semiparameric comparison of sochasic volailiy models using realised measures, Review of Economic Sudies 73, [15] Das, S.R., and R.K. Sundaram, 1999, Of smiles and smirks: A erm srucure perspecive, Journal of Financial and Quaniaive Analysis 34, [16] Demeerfi, K., E. Derman, M. Kamal and J. Zhou, 1999, More han you ever waned o know abou volailiy swaps, Goldman Sachs Quaniaive Sraegies Research Noes. [17] Dosis, G., D. Psychoyios, and G. Skiadopoulos, 2007, An empirical comparison of coninuous-ime models of implied volailiy indices, Journal of Banking and Finance 31, [18] Duan, J.-C., and C.-Y. Yeh, 2010, Jump and volailiy risk premiums implied by VIX, Journal of Economic Dynamics & Conrol 34, [19] Engle, R.F., and I. Ishida, 2002, Modeling variance of variance: The square-roo, he affine, and he CEV GARCH models, Working Paper, New York Universiy. [20] Eraker, B., 2004, Do sock prices and volailiy jump? Reconciling evidence from spo and opion prices, Journal of Finance 59, [21] Eraker, B., M. Johannes, and N. Polson, 2003, The role of jumps in reurns and volailiy, Journal of Finance 58, [22] Fukasawa, M., I. Ishida, N. Maghrebi, K. Oya, M. Ubukaa, and K. Yamazaki, 2010, Model-free implied volailiy: From surface o index, forhcoming in Inernaional Journal of Theoreical and Applied Finance. [23] Garcia, R., M.-A. Lewis, S. Pasorello, and É. Renaul, 2011, Esimaion of objecive and risk-neural disribuions based on momens of inegraed volailiy, Journal of Economerics 160, [24] Heson, S., 1993, A closed-form soluion for opions wih sochasic volailiy wih applicaions o bond and currency opions, Review of Financial Sudies 6, [25] Jacod, J., and V. Todorov, 2010, Do price and volailiy jump ogeher, Annals of Applied Probabiliy 20, [26] Jiang, G., and Y. Tian, 2005, The model-free implied volailiy and is informaion conen, Review of Financial Sudies 18,

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28 Appendix In his appendix, we derive he modificaions (36) and (37) of he relaions ha lead o a se of condiional momen condiions for GMM esimaion of he SV parameers o be applicable when each rading session lass less han 24 hours ( 1): E E V V H (40) E V a V b H (41) Var V A V B H (42) V V C 2 V D H (43) where E H is he expecaion under he physical measure, condiional on s H : V : s. These are, respecively, equaions (A.1), (A.2), (A.5) and (A.6) of BZ, which lead o: E V H E E V H H a V a b V H (44) 1 1 1E 1 Since G H, we obain, by he law of ieraed expecaions, E 1 1 1E 1, V V (45) which reduces o equaion (6) of BZ if 1 Furhermore, we have: ( 2 2 V H V H V H E Var E 2 A V B a V b A V B a V b 2a b V a V A 2a b V B b (45) 28

29 which is essenially equaion (A.7) in BZ, and E E V E V H H H E av 1 A 2 ab V 1 B b H 2 2 a V C V D A a b V B b E V H a C A a b V A 2a b a D 1 B b E V H a C a A 2b E H V a C a A b b A 2a b a D 1 B b E V I E V H J H (46) By he law of ieraed expecaions, we obain: E V E V I E V J (47) which reduces o equaion (10) in BZ if 1 29

30 30

31 Table 1 Mone Carlo Experimen Resuls Panel A True Parameer Se 1: (faser mean reversion) Bias 100 RMSE 100 GMM-BZ GMM-BZ-RL RL GMM-BZ Panel B True Parameer Se 2: (slower mean reversion) Bias 100 RMSE 100 GMM-BZ GMM-BZ-RL RL GMM-BZ

32 Table 2 Effecs of No Properly Adjusing he Momens for Marke Closure Bias 100 RMSE 100 Hours BZ RL BZ RL

33 Table 3 Realized Leverage as an Esimaor of under Heson ( 5 ) and Non-Heson CEV ( 10, 15 ) Bias 100 RMSE

34 Table 4 GMM Esimaion Using High-frequency S&P 500 and VIX h 24 h 65 GMM-BZ-RL (S.E.) (.0709) (.0687) (.1734) (.0095) (.0722) (.2424) (.3205) (.0095) GMM-BZ (S.E.) (.0708) (.0692) (.1746) (.0723) (.2413) (.3147) 34