Option Implied and Realised Measures of Variance

Transcription

1 Opion Implied and Realised Measures of Variance Damien Lynch and Nikolaos Panigirzoglou Firs Draf: January 3 This version: December 3 Moneary Insrumens and Markes Division, Bank of England, Threadneedle Sree, London ECR 8AH, damien.lynch@bankofengland.co.uk, nikolaos.panigirzoglou@bankofengland.co.uk. The views expressed in his paper are hose of he auhors and do no necessarily reflec hose of he Bank of England.

2 Absrac This paper analyses he differences beween forward (risk-neural expecaion) and realised variance. Four differen asses are examined: S&P 5, FTSE 1, eurodollar and shor serling fuures. Model-free measures of risk-neural forward variance are esimaed using he full cross secion of opion prices of a given mauriy. Inra-day fuures daa are used in he esimaion of realised variance. A mean squared error crierion is devised o inform he choice of opimal inra-day frequency. Marke microsrucure issues are considered for boh equiy and ineres rae fuures markes. The bias and efficiency of forward variance as a predicor of realised variance is reexamined. JEL Classificaion: G13; G19 Keywords: implied volailiy; realised volailiy; risk premia; marke microsrucure; high-frequency daa

3 1. Inroducion This paper examines he difference beween forward (risk-neural) and realised variance. Over a long period, his difference can be considered as an ex-pos measure of variance risk premia if marke paricipans are raional, ha is, hey don make any sysemaic errors in heir variance forecass. Mos of he pas lieraure has focused on he informaiveness and unbiasedness of implied volailiy from opion prices as a guide or predicor for he fuure volailiy of he underlying asse. A smaller par of he lieraure deal wih measuremen issues of boh forward (opion implied) and realised volailiy. We provide a brief descripion of his lieraure. Early sudies in he lieraure generally concluded ha, when compared wih pas volailiy, implied volailiy was a beer predicor of fuure volailiy. However some doub was raised as o wheher implied volailiy refleced all publicly available informaion a a given poin in ime. Laane and Rendleman (1976) using a cross secion of Black-Scholes implied volailiies for weny-four firms found ha he sandard deviaions implied in call prices were srongly correlaed wih acual sandard deviaion. Becker (1981) found, also using equiy opions, ha boh pas and implied volailiy were found o provide he bes forecas of fuure volailiy. Such a finding raises quesions abou he efficiency of implied volailiy as a forecas (does i fully reflec all informaion available o he marke of an opion conrac) and/or he accuracy of he Black-Scholes pricing model. This early empirical work was undermined by a lack of daa and small sample size. As more daa became available wih he expansion of opion markes, a number of papers sough o revisi he issue in he lae eighies and hroughou he nineies, focussing on ime series as opposed o cross-secional analysis. The papers by Day and Lewis (199), Canina and Figlewski (1993) and Lameroux and Lasrapes (1993) are noed for heir challenge o he conclusions of he earlier work wih findings ha refued he usefulness of implied volailiy as a guide o he fuure variabiliy of an asse price. Canina and Figlewski (1993) regressed daily one-monh volailiy for he S&P 1 index on observaions of one-monh implied volailiies on S&P 1 index opions and on pas S&P 1 index volailiy. The obvious overlapping observaion problem was addressed by adjusing he sandard errors. Implied volailiies from a 3

4 range of srike prices were examined raher han simply he a-he-money volailiy. Their regressions generally reurned very low R-squared and insignifican coefficien esimaes, promping hem o conclude ha implied volailiy is no a good predicor of fuure volailiy. Furher, hisorical volailiy was acually found o be a more accurae forecas han implied volailiy over heir sample period of Raher han provide a forecas of condiional volailiy, hey hus conclude ha implied volailiy is merely anoher piece of informaion o be used in is esimaion. Several sudies since he mid-199 s have ackled hese resuls from boh mehodological and economeric perspecives. This work has generally aken he line ha hese negaive conclusions are he resul of mis-measuremen of implied uncerainy; he risk neural naure of implied volailiy; inappropriae saisical inference (due mainly o overlapping observaions); and/or a wrongly specified pricing model. Jorion (1995) focussed on foreign exchange (FX) raher han equiy index opions. He claims ha biases in he measuremen of implied volailiies are less likely for FX opions as hese opions are raded side by side wih he underlying asse (hence no synchroniciy problems) and also because prices are rigorously checked and hus less suscepible o clerical errors. This could induce a sale price effec and also mean less scope for acive arbirage beween derivaive and spo markes. The implicaions of a poenial bid-ask disorion were also invesigaed. Having adjused for all of hese feaures, Jorion finds consisen evidence across hree currency pairs over he period ha implied volailiy conains informaion abou fuure volailiy and is a superior, hough biased, forecas. The bias in implied volailiy was found o be posiive so ha he fuure variabiliy of he underlying asse price implied by opions prices is, on average, greaer han ha subsequenly realised. These findings for FX implied volailiy have been suppored by ohers such as Xu and Taylor (1995). Chrisensen and Prabhala (1998) re-assess and grealy expand on he Canina and Figlewski (1993) paper. In paricular, hey use a much longer sample period allowing hem o use non-overlapping observaions. In addiion o he finding ha implied volailiy canno be ouperformed by purely hisorical-based forecass, hey also find ha, having adjused for a poenial errors-in-variables problem, he degree of bias in 4

5 implied volailiy forecass is much less han ha suggesed by previous empirical work in he lieraure. Fleming (1998) and Blair e al. (1) boh re-visi he issue for S&P 1 implied volailiy. In a very rigorous paper, Fleming (1998) aemps o minimise he errors in measuring implied volailiy for equiy indices and also employs a greaer degree of sophisicaion in his economeric and saisical analysis (providing GMM sandard error correcions for elescopic-fixed expiry dae mauriy paerns). Similar o Jorion (1995) he finds ha, hough biased, implied volailiy subsumes all informaion from hisorical volailiy as a forecas for fuure volailiy and ha he forecas errors from implied volailiy are no correlaed wih many variables ha are associaed wih condiional volailiy. These resuls were consisen across a range of forecas horizons from one day o several monhs and promp Fleming (1998) o conclude ha using a linear model ha adjuss for he bias in implied volailiy can provide a useful guide for fuure volailiy. Blair e al. (1) suppor hese findings in a paper ha conribues o he debae by using alernaive sources of daa for boh realised and implied volailiy. Realised volailiy is measured using inra-day daa (sampled every five minues) on he S&P 1 index while he CBOE Marke Volailiy Index (VIX) is used as a measure of implied volailiy.1 By consrucion he VIX avoids many of he poenial sources of measuremen error menioned earlier. Wih he excepion of he one-day-ahead horizon, Blair e al. (1) find ha S&P 1 implied volailiy can also ouperform very high frequency esimaes of hisorical volailiy in forecasing fuure volailiy. Bollen and Inder () also use high frequency inra-day daa o esimae daily realised volailiy. They accoun for heeroskedasiciy and auocorrelaion wih he VARHAC esimaor developed by den Haan and Levin (1996) by using a weak se of assumpions abou he daa generaing process. Focussing on he naure of he bias in implied volailiy as a measure of expeced fuure volailiy, some preliminary work by Chernov (1) heoreically jusifies, and seeks o explicily accoun for, he bias in implied volailiy by he exisence of a volailiy risk premium. He assumes a paricular sochasic process for acual 1 The VIX is a weighed average of S&P 1 opion implied volailiies ha is compued and made available on a real ime basis. By consrucion he index is a condiional volailiy measure ha is 5

6 volailiy so ha he esimaed ime-varying volailiy risk premium is model dependen. The volailiy risk premium is proxied by using hisorical volailiy. He finds ha implied volailiy is boh an unbiased and efficien forecas for fuure volailiy. However hese finding are undermined when broken down ino subsamples and his is aribued o economeric and measuremen difficulies. Overall a review of he lieraure seems o suppor a enaive conclusion ha implied volailiy does provide a beer forecas or guide for fuure volailiy han forecass based solely on hisorical informaion. As a forecas i is, however, biased and any aemp o use opion implied volailiies as a guide o fuure volailiy mus accoun for he volailiy risk premium. The analysis of his bias is he focus of his paper. We conribue o he exising lieraure in a number of ways. The firs conribuion relaes o he issue of measuremen of boh forward (risk-neural expecaion) and realised variance. In paricular, we use a measure of forward variance ha embodies informaion from he full cross-secion of opion prices on a given dae and is model independen. In his respec i is heoreically more appealing han measures such as he a-he-money volailiy or VIX index used in previous sudies. To measure realised variance we use high frequency daa. In choosing he opimal frequency we apply a mean squared error (MSE) crierion by combining he asympoic heory on realised variance wih an economerically robus esimae of he bias in realised variance due o marke microsrucural effecs. In addiion we examine equiy and ineres rae markes in boh he US and he UK, for a subsanial ime periods ha include evens such as he 1987 crash, he 199 ERM crisis, he 1997 Asian crisis and he 1998 Russian defaul and LTCM crisis. We provide evidence on he relaive imporance of microsrucural effecs in he differen markes and heir implicaions in erms of volailiy and volailiy risk premia. Finally, using hese measures of variance, we re-visi he predicabiliy of realised variance from forward variance implied from opion prices. In esing for efficiency we use, apar from an inra-day measure of hisorical variance, measures of marke-based, forward-looking and has a consan forecas horizon of one monh. For more informaion see Fleming (1998). 6

7 asymmery and faness of ails implied from opion prices. We provide esimaes of average, ex-pos, variance risk premia in boh equiy and ineres rae markes, assuming ha expecaional errors offse each oher over our relaively long sample periods. The paper is organised as follows. Secion provides discusses he esimaion and daa issues of forward variance. Secion 3 focuses on he measuremen of forward and realised variance. Secion 4 examines he bias and efficiency of forward variance as a predicor of fuure variance and analyses he properies of ex-pos variance risk premia. Secion 5 concludes.. Esimaion and daa issues for forward variance Opion premia for conracs wih differen srike prices can be used o provide a measure of uncerainy expeced by marke paricipans. Crucially, his measure reflecs he fuure volailiy ha would be expeced o occur in a world where invesors are risk neural. Insead real-world invesors are risk-averse and so care abou risk and a premium will be required by invesors o bear non-diversifiable volailiy risk. Brien-Jones and Neuberger () provide a characerisaion of all coninuous sochasic processes consisen wih a given se of opion prices, which enables riskneural forecass of variance o be obained ha are independen of he sochasic process of he asse price or volailiy. The only assumpion ha is needed is ha he sochasic process is coninuous. They derive he following risk-neural variance forecas: ds S S r ( T ) = P ( T, K) C ( T, K) e + dk (1) K S K T E where S is he underlying asse, r is he risk-free rae, C ( T, K), ( T K)are he P, European call and he pu prices respecively wih expiry dae a ime price K. T and srike 7

8 This variance forecas is relaed o he fair value of variance swaps ha were firs inroduced by Carr and Madan (1999) and Demeerfi, Derman and Kamal (1999). Variance swaps are forward conracs on annualised variance, ha is, he payoff depends on he difference beween he realised variance of he underlying asse over he life of he conrac and he delivery (forward) price for variance. Carr and Madan (1999) and Demeerfi, Derman and Kamal (1999) show ha he fair price of a variance swap (forward variance) is deermined by he cos of a replicaing porfolio given by he righ-hand side of equaion (1). This porfolio should consis of opion premia for conracs wih srikes ranging beween zero and infiniy. Therefore, alhough he derivaion of equaion (1) is independen of he sochasic process of he underlying asse, i assumes he exisence of a coninuum of srikes, which in pracice doesn exis. This means ha he forward variance of equaion (1) depends on he inerpolaion mehod used o generae a coninuum of srikes from a limied range of available srikes. This limiaion increases he risk of hedging variance swaps. The variance forecas in equaion (1) differs from Black-Scholes implied variance in ha i conains informaion from all raded srikes and no jus he a-he-money. More imporanly, he Black-Scholes model assumes consan volailiy and, as noed in Brien-Jones and Neuberger (), i is inconsisen o forecas volailiy from consan volailiy models. Addiionally, in conras o VIX or similar measures, i is more heoreically appealing in ha i does no rely on ad hoc weighs or resricive model assumpions. Anoher imporan issue menioned in Brien-Jones and Neuberger () is he convexiy adjusmen bias when forecasing volailiy insead of variance as implied by equaion (1). In paricular, he risk-neural forecas of volailiy, ha is, he riskneural forecas of he square roo of variance E T ds S will be smaller han he S square roo of he forward variance e r ( T ) P ( T, K) C ( T, K) + dk. In K S K Appendix 1 we derive a formula for he convexiy adjusmen. This is an imporan issue for he analysis of he differences beween forward and realised uncerainy in 8

9 secion 4. Alhough Table 1.1 shows ha he average convexiy adjusmen is no very large (beween.15-.3%), he adjusmen, iself a funcion of volailiy, is ime varying and so regressions in volailiy space are likely o be affeced. For his reason we compare realised and forward uncerainies in variance space as opposed o volailiy (i.e. sandard deviaion) space. As menioned above, o esimae he risk-neural forecas of variance, ha is, he righ-hand side of equaion (1), we need a coninuum of srikes. To generae a coninuum of srikes we employ he cubic smoohing spline inerpolaion of he implied volailiy smile in dela space, as described in Bliss and Panigirzoglou (). Afer fiing 3 he smoohed implied volailiy funcion, poins along he curve are convered ino European call and pu prices (using he Black-Scholes formula). These opion prices are hen used in he esimaion of forward variance: S e r ( T ) P ( T, K) C ( T, K) + dk. K S K The opion conracs used in his paper have fixed expiry daes, ha is, he ime o mauriy changes wih ime. Our inpu daase includes daily call and pu opion and fuures prices on all raded (quarerly) conracs for he FTSE 1 index and S&P 5 fuures opions and hree-monh serling (shor-serling) and eurodollar LIBOR ineres rae fuures opions. The FTSE 1 and shor serling opions conracs are all raded on London Inernaional Fuures and Opions Exchange (LIFFE). Daily selemen prices from LIFFE were obained from 1987 for shor serling fuures opions and from 199 for FTSE 1 index opions. The associaed value of he underlying was he fuures price repored by LIFFE. Opions conracs on eurodollar fuures and he S&P 5 fuures are raded on he Chicago Mercanile Exchange and daily selemen prices for hese opion conracs were obained from 1983 for S&P 5 and from 1985 for eurodollar. The associaed value of he underlying was he selemen price of he fuures conrac mauring on or jus afer he opion expiry The inerpolaion of he implied volailiy smile in dela space has he advanage ha far ou-of-hemoney opions are grouped ogeher in he ails allowing for more shape near he cenre of he disribuion where more rading occurs. 3 The smoohing parameer conrols he radeoff beween smoohness and goodness-of-fi. Afer experimening wih differen values we chose a value of.99 for all conracs used in his sudy. 9

10 dae. The risk free raes used are he Briish Bankers Associaions 11am fixings for shor serling and eurodollar LIBOR raes repored by Bloomberg. Only a- and ou-of-he-money call and pu prices were used because here is usually more rading in hese, raher han, in-he-money opions (see Bliss and Panigirzoglou ()). Opion prices ha violaed he monooniciy 4 or convexiy 5 properies were discarded. Opion prices for which an implied volailiy was impossible o compue or wih delas smaller han.1 or greaer han.99 (far ou-of he money opions wih usually lile or no rading) were also discarded. Following his screening process, we fi he available implied volailiies of a given mauriy cross secion in dela space wih he smoohing spline mehod. However, i is necessary o exrapolae he spline beyond he range of available daa. Since he spline mehod exrapolaes linearly ouside he available range (resuling someimes in negaive or implausible large implied volailiies) we force he spline o exrapolae horizonally. This is done by inroducing hree pseudo srikes above and below he available range wih implied volailiies equal o ha of he respecive available exreme srikes. The oupu from he esimaion process consiss of daily forward variances for all available expiry daes. To compare hese measures of forward variance wih hose from he Black-Scholes model in volailiy space, we adjus he square roo of he forward variance o accoun for convexiy and examine he difference beween his measure and he Black-Scholes a-he-money volailiy. 6 In calculaing he convexiy adjusmen we use he convexiy adjusmen formula se ou in Appendix 1, where E [ y] is he forward variance (i.e. ha of equaion (1) above) and Var y is proxied wih he variance of he forward variance, using a hree-monh rolling window. These differences are ploed in Appendix and he average absolue differences are shown in Table.1. On average he difference beween he wo measures is around 1.5 percenage poins and hus can have a significan effec when examining he bias in risk-neural or forward volailiy as a predicor of realised volailiy. [ ] 4 Monooniciy requires ha he call (pu) prices are sricly decreasing (increasing) wih respec o he exercise price. 5 I requires ha a buerfly spread a a paricular srike (formed by selling wo call opions a his srike and buying he wo adjacen call opions) is posiive. In a coninuum of srikes, his is equivalen o require ha he call and pu price funcions are convex. 1

11 3. Esimaion and daa issues of realised variance Much of he recen lieraure on finance has focussed on he accurae esimaion of he volailiy of asse prices. Volailiy is usually considered o be unobservable, a corollary o he view ha daa generaing processes for asse prices are laen in naure. However compuaional and heoreical advances (diffusions ec.), as well as he availabiliy of ransacion by ransacion asse prices, have brough us closer o an unbiased, consisen and efficien esimaor for volailiy. The esimaor for realised volailiy ha is uilised in mos sudies is very simple. I is based on evaluaing he quadraic variaion of he price series ha is, a sum of he squared asse price reurns (SSR) a a paricular frequency over a paricular horizon. This provides an esimae of he inegraed volailiy of he underlying asse price process; he acual volailiy for a period wihin his horizon is hen aken as he difference beween successive inegraed volailiies. Recen advances in he undersanding of he properies of quadraic variaion as an esimaor of inegraed volailiy have found i o be a consisen and efficien esimaor (Barndorff-Nielsen and Shephard ()). In finie/smaller samples, Mone Carlo experimens sugges ha a high number of inra-day observaions are needed o rely on he asympoic resuls (Barndorff-Nielsen and Shephard (3). These resuls have led some papers in he lieraure ha use inra-day daa o measure volailiy in his way, o regard he resuling esimaed volailiy series o be as near o he rue volailiy of he asse price process as o regard rue volailiy as observable (see for example Andersen e al ()). In regarding volailiy as close o observable, i may be examined wihou he need o accoun for measuremen error. Recen addiions o he lieraure have sough o use inra-day daa o provide more reliable esimaes of realised volailiy (for example see Taylor and Xu (1997), Andersen e al (, 1), Bollen and Inder ()). Some of hese papers have used he sums-of-squares esimaor menioned above. However he use of his esimaor wih very high frequency daa can be problemaic. The lieraure on marke microsrucure has idenified several feaures ha are observed empirically in high 6 The Black-Scholes implied volailiy can be regarded as an expecaion of fuure volailiy for near- 11

12 frequency daa. These can induce auocorrelaion a very high frequencies (e.g. 1- minue or 5-minue). Chief among such feaures is he impac of he bid-ask spread, infrequen rading, order-spreading ec. Use of he SSR esimaor requires serially uncorrelaed daa. Andersen e al. () specifically guard agains he indiscriminae use of he SSR in hese circumsances. They explicily recognise a radeoff in choosing a high enough frequency o produce an esimae wih minimal sampling variaion on he one hand, bu avoiding he biases ha may be induced by microsrucural feaures in very high frequency daa on he oher. In choosing he opimal frequency hey propose a ool called he volailiy signaure plo. Choice of frequency involves visual inspecion of a plo of realised volailiies over various inra-day frequencies. The frequency a which he plo flaens ou i.e. produces successively similar volailiies is aken o be he opimal frequency. Bollen and Inder () revisi he heoreical basis for using he SSR esimaor in measuring realised volailiy and seek o ake accoun of he naure of hese empirical feaures. They adap a procedure firs developed by den Haan and Levin (1996) ha allows consisen esimaion of covariance marices in vecor auoregressions in he presence of boh heeroscedasic and auocorrelaed errors. Bollen and Inder () show his esimaor, known as he VARHAC esimaor (described below), o be well suied o he high frequency reurn daa ha we are ineresed in. Theoreically, hey provide an accurae esimae of realised volailiy ha is no confounded by microsrucural effecs, a he highes frequency available. This paper seeks o conribue o he exising lieraure by also uilising inra-day daa in he esimaion of realised volailiy. The mehodologies se ou above o esimae realised volailiy, hough inuiively appealing, can involve eiher some subjecive choice for he praciioner (Andersen e al () case), or are compuaionally inensive, and hus impracical, over large samples (Bollen and Inder ()). Our approach is o combine hese wo mehodologies wih he recenly developed asympoic heory on realised variaion by Barndorff-Nielsen and Shephard (, 3). In doing so we devise a framework ha will provide for an objecive choice of inra-day frequency a which o apply he SSR esimaor o produce a measure of realised volailiy ha is robus (economerically) o unwaned microsrucural effecs. expiraion, a-he-money opions. See Feinsein (1989) for furher deails. 1

13 The core of our approach o choosing an opimal inra-day frequency lies in a mean square error (MSE) crierion. For any esimaor, he MSE may be decomposed as he sum of he square of he bias of he esimaor and he squared sandard error of he esimaor. Employing his decomposiion, we seek o combine, for each frequency, he bias in he SSR esimaor, due o microsrucural effecs, wih an esimae of he sandard error of he SSR esimaor. In doing so, we explicily recognise boh elemens of he radeoff - he auocorrelaion bias and he reducion in sandard error due o larger samples a higher frequencies - involved in choosing an opimal inraday frequency. We hen choose ha frequency ha is found o have he minimum MSE, hus minimising he radeoff. 3.1 Esimaing he serial correlaion bias In arriving a an esimae of he bias in he SSR esimaor of realised variance, we compare SSR esimaes of realised variance wih he consisen variance esimaes obained by implemening he esimaion echniques se ou in Bollen and Inder (). The mehodology developed in ha paper is based on applying he vecor auoregressive heeroskedasic and auocorrelaion consisen (VARHAC) esimae of he variance/covariance marix of a daa process as se ou in den Haan and Levin (1996). Bollen and Inder () find ha he relaively weak condiions for he use of he VARHAC esimaor are well saisfied by inra-day daa. In he case of a univariae analysis, he core of he VARHAC esimaor involves esimaion of an auoregression of he squared inra-day reurns. The residuals from his auoregression are hen used o calculae he specral densiy of he reurns. Specral densiies are commonly used in ime series analysis o sudy cycles in ime series (a various rigonomeric, as opposed o ime, frequencies) and are closely relaed o he auocovariance funcion of a ime series. Evaluaion of he specral densiy a a rigonomeric frequency of zero provides an esimae of he variance of he ime series. Thus, Bollen and Inder (), following den Haan and Levin (1996), evaluae he specral densiy of he reurns, obained using he residuals from he auoregression, a a rigonomeric frequency of zero o esimae he realised variance using inra-day frequency. The use of he residuals from he auoregression means 13

14 ha he daa used in esimaing he realised variance of he series has been cleansed of he effecs of any serial correlaion ha may be presen. Implemenaion of he VARHAC esimaor is exremely inensive. Auoregressions are repeaedly run for a given ime frequency, o esimae he daily realised variaion. In choosing he order of he auoregression, a esing down procedure where, beginning wih a maximum lag, auoregressions a successively lower orders are esimaed. The choice of lag lengh for each frequency, each day, is based on maximising an informaion crierion, such as he Akaike (1973) Informaion Crierion (AIC) or Schwarz (1978) Bayes Informaion Crierion (BIC). This procedure mus be repeaed each day for each frequency and involves he esimaion of a very subsanial number of regressions. For our purposes comparison of he realised wih implied variance involves esimaing realised variances over, very ofen, much longer periods of ime han jus one day. The ime o expiry of an opion, and so he horizon for he forward variance ha we calculae, declines each day. This horizon may be as grea as six monhs and as low as five days for example. As a resul implemenaion of his sraegy wih samples of inra-day daa over periods ranging from eleven o weny years is no feasible. To address his, we obain esimaes of realised variance using he VARHAC procedure a a se of fixed numbers of days o mauriy for each conrac, raher han a every rading day. The process can be summarised as follows: 1. We focus on en inra-day frequencies 1, 5, 1, 15, 3, 45, 6,1 and 4 minues. The 4 minue frequency provides one observaion per rading day (i.e. he las rade).. We esimae boh SSR and VARHAC realised variance, a each of frequencies, over he remaining lives of he opion conracs on rading days ha are approximaely 3, 6, 9, 1 and 15 days away from expiry. 3. For each conrac, he average difference beween he VARHAC and SSR variances is hen aken o provide an esimae of he bias in he SSR esimae. In his way, he VARHAC esimaed variance is aken o be a proxy for he rue realised variance of he asse concerned. 14

15 Implemenaion of he VARHAC esimaor follows he procedure se ou in Bollen and Inder () and den-haan and Levin (1996) and so is jus summarised in he following seps: a) For a specified frequency and a given conrac wih ime o mauriy T, we esimae he following regression for each day = T-3x; x=1,,..,5, using all of he inra day observaions, j=1,,j(t-3x), where J is he oal number of inra-day observaions a he specified frequency beween and T: ~ K r j, = β i r j, i + e j,, for each lag lengh k = 1,, K, K. i= 1 In line wih he advice in den Haan and Levin (1996) we ake ~ K o be ~ ( J ( T 3x)) 1/ 3. b) Again following den Haan and Levin (1996) we compue BIC values for each lag lengh k and dae. For each day, we hen selec he residuals from he regression ha minimises he BIC across he differen values of k. We denoe he opimal value of k on dae, k he corresponding residuals, e, where e is a ( J ( T 3x)) 1vecor and he k 1vecor of coefficiens β. c) The VARHAC realised variance is hen esimaed as: ^ ^ ^ ^ σ VH, T = 1 ^ ^ K k = 1 ^ e ' e ^ β k, 3. Esimaing he sandard error of he SSR realised variances The second elemen of he MSE relaes o he accuracy wih which he SSR esimaor can esimae realised variance. Theoreically, he more daa ha is used he greaer he accuracy, and he one minue frequency should provide as close an esimae as possible of acual realised variance. The gain in erms of accuracy mus be measured relaive o he loss in erms of bias a he very high frequencies. To capure he 15

16 sandard error of he realised variances esimaed using he SSR esimaor, we use he relaively recen lieraure on he asympoic heory of realised variaion. In he conex of a fairly general sochasic volailiy model, Barndorff-Nielsen and Shephard () develop a limi heory for he difference beween realised and inegraed volailiy, where inegraed volailiy refers o he coninuous ime analogue of sums of squared reurns and hus refers o acual volailiy in a heoreical sense. This limi heory provides us wih informaion on he asympoic properies of he SSR esimaor. One of he properies of he SSR esimaor, under his heory and assuming a very general sochasic volailiy model, is ha i is a consisen esimaor for acual realised variance. Tha is, Barndorff-Nielsen and Shephard () show ha he SSR esimaor converges asympoically o a normal disribuion, wih a mean given by he acual volailiy. However, he usefulness of his resul in finie samples is quesioned in Barndorff-Nielsen and Shephard (3), where hey use Mone Carlo mehods o show ha very high inra-day frequencies are needed for a good approximaion in finie samples. They also find ha he approximaion in finie samples improves when a ransformaion o logarihmic space is used (ha is, he SSR esimaor is lognormally disribued), even for lower inra-day frequencies. Tha volailiy is beer approximaed in finie samples by a lognormal disribuion is a resul ha is acknowledged in he lieraure. Andersen e al (1) find ha in he case of equiy volailiies, logarihmic sandard deviaions for 3 socks from he Dow Jones Indusrial Average are well approximaed by a normal disribuion. In he case of ineres rae implied volailiies, Lynch and Panigirzoglou (3) also find logarihmic implied volailiies o have normal disribuions over samples of a leas en years of daa. Following Barndorff-Nielsen and Shephard (3), we have ha ( log SSR logσ ) 3 n j= 1 M M j= 1 y y 4 j, n j, n n L N(, 1), () 16

17 denoing acual volailiy over horizon n by σ, he SSR esimaor by SSRn and n y n, j as he jh period reurn of M inra-day periods. The denominaor of () provides an esimae of he sandard error of he logarihm of he SSR esimaor. To esimae he sandard error for he SSR esimaor iself using his resul we make he following ransformaion from he normal variable log SSRn o a lognormal variable SSR n : Var (log SSRn ) ( SSR ) = { E( SSR )}.( e 1 Var ). (3) n n Proxying E(SSR n ) wih M j= 1 y j, n and using he square of denominaor from () as he variance of log SSRn, we esimae he sandard error of he SSR esimaor as: M 4. y M j, n j= 1 Sd. error ( SSR ) n = y j, n. exp 1. (4) j= 1 M 3. y j, n j= Implemening he mean square error crierion and choice of frequency Our daase includes ransacion-by-ransacion fuures prices for he S&P 5 (1983-1) and FTSE 1 (199-1) equiy indices and for shor serling (1988-1) and eurodollar (1985-1) ineres rae fuures. Trading hours for he exchanges concerned varied during he samples. In he US, he Chicago Mercanile Exchange (CME) S&P 5 fuures raded beween 8:3 15:3, and eurodollar fuures beween 7: 14:. In he UK, rading hours a London Inernaional Fuures and Opions Exchange (LIFFE) for FTSE 1 fuures were 8:3-17:3 for pre-sepember 1999 period and 8:-17:3 in he pos-1999 period; while shor serling fuures raded from 8:-16: up o March 199, beween 8:-18: o March 199, 8:-18: o Ocober 1991 and finally changing o 7:3-18: in July 1. The raw daa obained from LIFFE (UK) and he Insiue for Financial Markes (US) were run hrough filers o clean he daa and remove implausibly large observaions due o 17

18 errors in he recording of he ransacion daa. Time series of prices a each inra-day frequency, beween he relevan exchange opening and closing imes, for each asse were hen consruced by sampling he filered daa a each frequency every rading day beween a given rade dae and he corresponding opion expiry dae. The reurn series were hen obained by aking he firs difference of he logarihm of he demeaned price series a each frequency, for a given number of days o he expiry dae. Average realised variances using he SSR and VARHAC esimaors, across all en inra-day frequencies, are ploed in Chars A.1-4 for he four asses. These averages are based on a rimmed sample whereby 5% - ha is, upper and lower.5% - of he oal number of observaions were disregarded. The profiles of SSR realised variances over he inra-day frequencies clearly indicae he imporance of microsrucural effecs in esimaing realised volailiy in very high frequency daa. For almos all asses, he SSR esimaor produces he highes esimaes of realised variance a he highes 1-minue frequency wih subsequen esimaes declining as he frequency decreases. I is clear from hese chars ha esimaing realised variance by arbirarily selecing an inra-day frequency is illadvised. The profile of mos SSR variance esimaes appears o flaen a around 45-6 minues. The rae of decline is paricularly marked in he case of shor serling and eurodollar, relaive o ha for he wo equiy indices. The declining paerns of SSR realised variance for shor serling and eurodollar reurns suggess greaer degrees of negaive serial correlaion as frequency increases. This finding is consisen wih wha we would expec o see for a highly liquid asse where negaive serial correlaion could be induced by he bid-ask bounce, as posied by Roll (1984). The shape of he equiy index SSR variance profiles over various frequencies suggess ha fuures reurns on hese indices may be less affeced, on average, by his kind of negaive serial correlaion. I is possible o posi a number of poenial explanaions for why his may be so. Equiy index fuures are less raded han ineres rae fuures so ha he bid ask-bounce effec could be weaker. 7 In addiion, fuures on equiy 7 Evidence abou he relaive urnovers of equiy and ineres rae fuures conracs can be found in BIS Quarerly Review, June 3. For example, for, equiy fuures urnover was approximaely 5% of urnover in he ineres rae fuures. 18

19 indices may be subjec o oher ypes of microsrucural feaures. For example, equiy indices may suffer from a non-synchronous rading problem whereby a marke-wide shock is no refleced in all individual consiuen equiy prices simulaneously. As a resul, he shock may impac on he index value, which is he weighed-value of he consiuen prices, on a more gradual basis as individual consiuen equiies are subsequenly raded. The working of orders in financial markes could also serve o induce a posiive serial correlaion in equiy index reurns. In his case, large orders canno be filled in one aemp as marke makers can specify a volume limi wih each bid/ask quoe. This means ha such orders mus be worked hrough he marke, saring wih he bes quoe and coninuing unil he order is filled. I is no likely ha his would be compleed simulaneously so ha he price informaion for he index in he large order is refleced gradually and no insananeously. Thus he order working and non-synchronous effecs may well be inducing a degree of posiive serial correlaion in equiy index reurns ha could poenially offse, parially a leas, any induced negaive serial correlaion due o a bid-ask bounce effec. Turning o he esimaed MSEs o formally choose a frequency for each asse, he esimaes of he average squared biases, squared sandard errors and corresponding MSEs for he SSR esimaors for all four asses are presened in Appendix 3, Chars A As before, rimmed means are shown. The chars highligh he relaive conribuions of he wo componens in he calculaion of he MSE crieria. For mos asses, he squared serial correlaion bias is of greaer, or a leas equal, magniude o ha of he squared sandard error of he SSR esimaor. As a resul he gain, in erms of accuracy of esimaion of realised variance, from using inra-day day a increasingly higher frequencies is usually mached or ouweighed by he biases ha are incurred. Ineresingly, he bias profiles, and heir relaive weigh in he MSE crieria, differ subsanially beween equiy index and ineres rae fuures. As alluded o in he comparison beween he average VARHAC and SSR esimaes of realised variance in sub-secion 4.1 above, he serial correlaion biases for he ineres rae fuures SSR esimaes are much greaer han hose for he wo equiy indices. SSR esimaes for eurodollar fuures were found o have he highes squared biases among all of he asses, exceeding hose of he FTSE 1 and S&P 5 by a facor of 19

20 up o 16. Generally he ineres rae biases were found o be greaes a he 1 minue frequency, declining ou o around 45 or 6 minues. For all frequencies, he ineres rae biases dominae he esimaed MSEs, hus ruling ou very high-frequency daa in erms of esimaing realised variance using he SSR esimaor. The minimum MSE crieria insead sugges frequencies of 6 minues for shor serling fuures, and 45 minues for eurodollar fuures. For equiy index fuures, he greaes biases occurred a he 4 minue or daily frequencies, hough he FTSE 1 also exhibied a srong bias a one minue inervals. In conras o he ineres rae fuures, he equiy index squared biases and sandard errors were similar in size, suggesing ha marke microsrucure effecs are much weaker in equiy index fuures inra-day daa. The equiy index fuures squared biases are relaively fla beween he daily and ulra-high frequencies and he esimaed MSE s are minimised a 45 minues for FTSE 1 and 3 minues for S&P 5. Using he chosen frequencies for he four asses, he SSR esimaor was hen applied o all rading days for all conracs o produce a series ha maches ha of he forward variance series in erms of ime o mauriy each rading day, over he enire sample, for each asse. I is he difference beween hese wo he mached forward and realised variances ha is he main subjec of he analysis in he remainder of he paper. 4 Predicing realised wih forward variance In his secion we consider he informaion conen of forward for realised variance. Our analyses are conduced by examining coefficiens, and heir saisical significance, from OLS regressions. However, here are wo feaures in our daa ha we mus firs address. Our variables consis of variances, esimaed each day, over a horizon ha is deermined by he number of days o a fixed opion conrac expiry dae. As a resul, he variance horizon declines over successive days, evenually reaching zero a he expiry dae. Thus variances esimaed for a given conrac are overlapping and so highly serially correlaed. Moreover, as poined ou by Fleming (1998), he daa ha

21 we use is elescopic in naure. The elescopic feaure arises as insead of looking ahead for a similar fixed period of ime, he daa are overlapping for he life of an opion conrac only. As a resul, here is no overlap among variance observaions across conracs. The presence of hese feaures in he daa means ha a simple applicaion of OLS would yield inefficien coefficien esimaes, wih inaccurae sandard errors, hus undermining saisical inference. Given he elescopic naure of he daa, use of he Newey-Wes heeroskedasic and auocorrelaion consisen (HAC) mehod of covariance marix esimaion is no appropriae. To deal wih his, Fleming (1998) develops consisen generalised mehod of momens (GMM) esimaors ha can produce variance-covariance marix esimaes ha are robus o residual serial correlaion and condiional heeroskedasiciy in he presence of elescoping observaions. The Fleming (1998) covariance esimaor is an exension of exising HAC mehods. The Newey Wes (1987) HAC covariance esimaor generalises he regular leas squares covariance marix esimaor o allow for serially correlaed errors. Exra cross produc erms are included in he calculaion of he covariance marix by weighing observaions wih he auocovariances of he residuals. 8 The number of exra cross produc erms included is, in heory, deermined by he order of auocorrelaion of he residuals. The HAC covariance marix esimaor developed by Fleming (1998) limis he auocovariance erms o include only hose observaions ha are common o an individual conrac. Using his idea we modify he Newey Wes HAC covariance esimaor o esimae he following HAC covariance marix wih elescopic observaions: Ω = 1 N N ˆ ε. x x + = 1 k = 1 i= k + 1 L N φ ˆ ε. ˆ ε k i ( x x + ˆ ε. x x ), i i where, εˆ are he OLS residuals, x is he -h row of a regressor marix, N is he oal number of observaions, L is he order of residual auocorrelaion (aken o be 9 days) wihin conracs and φ k akes a value of one if he -h and (-i)-h observaions relae o he same conrac, and a value of zero oherwise. So φ k acs o limi he 8 In addiion o he auocovariances, he weighs on he observaions also include anoher facor o ensure a posiive semi-define covariance marix. 1

22 overlap in observaions o hose wihin conracs only. As a resul we can esimae a covariance marix ha is robus o he high serial correlaion wihin conracs and so adjus for he elescopic naure of he daa. In reporing he sandard errors for our coefficien esimaes in laer regressions, we show boh he Newey Wes sandard errors and hese elescopic Newey Wes sandard errors. We begin our comparison of realised variance wih forward opion-implied variance by esimaing radiional regressions of realised variance (annualised) on a consan and forward variance (annualised). If forward variance is an unbiased expecaion of realised variance, hen he consan should be close o zero and he slope coefficien on forward variance close o one. To assess his join hypohesis, Wald ess are applied wih he appropriae resricions on he coefficiens. The resuls of he regressions and saisical es are presened in Table 4.. These show ha for all asses and all sample periods he consan is close o zero bu he slope coefficien is significanly differen o one (a 1% significance levels) generally of magniude Moreover, he Wald ess fail o accep he join hypoheses in all cases, confirming he finding in he previous lieraure ha forward variance is a biased expecaion of fuure realised variance. Furher, wih all of he slope coefficiens less han one, his suggess ha forward variance, on average, overesimaes subsequenly realised variance. Noneheless, forward variance does possess some informaion abou fuure realised variance wih he adjused R-squared suggesing ha i can explain beween -48% of he variaion in realised variance. To provide an ex-pos esimae of he average bias in forward variance, Table 4.1 shows he average differences beween he forward and realised variances, for each of he four asses wih varying sample periods, ogeher wih heir sandard errors (Newey Wes and elescopic Newey Wes adjused). For he FTSE 1 and S&P 5 equiy indices, mos of he differences are of similar magniude; are negaive and highly significan wih probabiliy values of one percen or less. For he full sample for he S&P 5 we were, however, unable o rejec he hypohesis ha he difference beween realised and forward variances is equal o zero. Tha his sample (1983-1) conrass wih laer S&P 5 samples ( for example) is likely o be due o he large influence of he 1987 Crash. As he Crash was no anicipaed by financial markes, realised variance exceeded ha which would have been expeced o

23 occur. I is his subsanial underpredicion of variance, in conras o he usual overpredicion ha resuls in he failure o rejec he null hypohesis of zero mean difference for he full S&P 5 sample. Turning o ineres raes, he average differences are again mosly negaive. As for he equiy indices, forward variance was found, on average, o exceed subsequenly realised variance. Only in he period beween 1988 and 1, was he difference found o be saisically insignifican for shor serling. This period includes he ERM crisis and, as wih he S&P 5 and he 1987 Crash, here was a large underpredicion of volailiy a ha ime. The averages repored in Table 4.1(a), for boh equiies and ineres raes, could be considered as ex-pos measures of annualised variance risk premia, if expecaional errors average ou over he sample periods. The size of he (annualised) variance risk premium for ineres raes (when i is significan) is abou.5, half of ha for equiies (.1). Mapping hese numbers for he variance risk premium ino volailiy (i.e. sandard deviaion) space is no sraighforward. In doing so one has o ake ino accoun convexiy adjusmens which hemselves, as funcions of variance, are unlikely o be consan. To provide an indicaion of he poenial magniudes of he risk premia in volailiy space we obain an esimae of forward volailiy by applying an adjusmen o he square roo of forward variance for convexiy, as explained on page 1. Table 4.1(b) shows he mean differences beween hese convexiy-adjused forward volailiies and he corresponding mauriy-mached realised volailiy. Again, as for he differences in variance space, he differences are all negaive and, exceping he full sample for shor serling which includes he ERM crisis period, saisically significan. The magniude of he ex-pos volailiy risk premia range from -3% for equiy indices o 1-% for ineres raes. The difference in he sizes of he volailiy risk premia beween equiy indices and ineres raes may reflec higher equiy index volailiy, on average. For example, he average forward volailiy over he sample for he S&P 5 is 18% compared o 15% for eurodollar. 3

24 Comparing hese risk premia esimaes wih he differences beween he convexiyadjused forward volailiy and a Black-Scholes measure of implied volailiy in Table.1, suggess ha he use of Black-Scholes volailiy would imply much lower volailiy risk premia. This highlighs he imporance of measuring forward variance accuraely using informaion from he whole cross-secion of opion prices as opposed o jus he a-he-money conrac price. Nex, we consider he efficiency of forward variance in predicing fuure realised variance by including a measure of backward-looking variance in our regressions. Some previous sudies, such as Chernov (1), also used hisorical variance as a proxy for he poenial bias due o volailiy risk premia. Table 4.3 shows resuls from an exension of he regressions of realised on forward variance o include backward variance. The measure of backward variance is based on sums of squared hisorical inra-day reurns of he same frequency as ha used for realised variance. As wih realised variance, he backward variance is esimaed over a period ha maches he horizon of he forward variance, ha is, he ime o expiry of he corresponding opion conrac. So wih T days o expiry, backward variance is calculaed using squared inra-day reurns from he mos recen T rading days, i.e. hose days ha are beween T and T+1 days o expiry. The resuls show ha backward variance is saisically insignifican for all asses and sample periods. This is consisen wih he more recen lieraure (e.g. Fleming (1998), Chrisensen and Prabhala (1998), Blair e al. (1)). A corollary of our resuls is ha he backward variance is unlikely o ac as a proxy for variance risk premium. Finally, in a furher es of efficiency, we use some addiional informaion from opion prices. In paricular, we use a measure of asymmery (proxied by sandardised risk reversal) and faness of ails (proxied by sandardised srangle). 9, 1 Table 4.4 shows 9 Sandardised risk reversal is given by he difference beween he 5-dela call and 75-dela call implied volailiies, divided by he am (5-dela) volailiy. I reflecs he slope of he volailiy smile. Since i is divided by he am volailiy i adjuss for changes in uncerainy. A lognormal pdf has a risk reversal equal o zero, ha is, he risk reversal shows he asymmery of he implied pdf in excess of he benchmark lognormal pdf. 4

25 ha here is lile value o be added o forward variance in predicing realised variance by using he risk reversal and srangle and backward variance as addiional variables (Table 4.4). As a resul, he bias in forward variance is unlikely o be relaed o hisorical variance, asymmery or fa ails in marke expecaions. Overall, he resuls in Tables show ha we canno rejec he hypohesis ha forward variance is an efficien forecas of realised variance. 5 Conclusions The issue of he relaion beween realised and forward (i.e. risk-neural expecaion) of variance has been exensively sudied in he lieraure. The resuls have been mixed depending on he measures used, asses examined, sample periods covered and economeric echniques employed. We add o his lieraure by using a heoreically more consisen measure of forward variance; a measure of realised variance based on high frequency daa; and by re-examining he bias and efficiency of forward as a predicor of realised variance. We exploi he resul of Brien-Jones and Neuberger () o obain a measure of risk-neural expeced variance based a saic rading posiion in European call and pu opions. This measure is model independen in ha i does no require any assumpions o be made abou he dynamics of he underlying asse or is volailiy. As such i is a superior measure o he Black-Scholes implied volailiy ypically used in previous sudies. In addiion, i is also a more heoreically consisen measure han he VIX, which uses an ad hoc weighing scheme o combine implied volailiies of differen srikes or mauriies. High frequency inra-day daa can improve he measuremen of realised volailiy by increasing is efficiency. However, microsrucural effecs a high frequencies can induce auocorrelaions ha creae biases in he sum of squared reurns (SSR) 1 Sandardised srangle is given by he difference beween he average of he 5- and 75-dela call implied volailiies and he am volailiy, divided by he am volailiy. I provides a measure of he degree of curvaure of he volailiy smile. Since i is divided by he am volailiy i adjuss for changes in uncerainy. A lognormal pdf has a srangle equal o zero, ha is, he srangle shows he degree of faness of ails of he implied pdf in excess of he benchmark lognormal pdf. 5