Implied Volatility from Options on Gold Futures: Do Econometric Forecasts Add Value or Simply Paint the Lilly?

Transcription

1 WORKING PAPER SERIES Implied Volailiy from Opions on Gold Fuures: Do Economeric Forecass Add Value or Simply Pain he Lilly? Chrisopher J. Neely Working Paper C hp://research.slouisfed.org/wp/003/ pdf July 003 Revised June 004 FEDERAL RESERVE BANK OF S. LOUIS Research Division 4 Locus Sree S. Louis MO 630 he views expressed are hose of he individual auhors and do no necessarily reflec official posiions of he Federal Reserve Bank of S. Louis he Federal Reserve Sysem or he Board of Governors. Federal Reserve Bank of S. Louis Working Papers are preliminary maerials circulaed o simulae discussion and criical commen. References in publicaions o Federal Reserve Bank of S. Louis Working Papers (oher han an acknowledgmen ha he wrier has had access o unpublished maerial) should be cleared wih he auhor or auhors. Phoo couresy of he Gaeway Arch S. Louis MO.

2 Implied Volailiy from Opions on Gold Fuures: Do Economeric Forecass Add Value or Simply Pain he Lilly? Chrisopher J. Neely* June 004 o be possess d wih double pomp o guard a ile ha was rich before o gild refined gold o pain he lilly o hrow perfume on he viole o smooh he ice or add anoher hue Uno he Rainbow or wih aper ligh o seek he beaueous eye of heaven o garnish Is waseful and ridiculous excess. William Shakespeare King John. Absrac: Consisen wih findings in oher markes implied volailiy is a biased predicor of he realized volailiy of gold fuures. No exising explanaion including a price of volailiy risk can compleely explain he bias bu much of his apparen bias can be explained by persisence and esimaion error in implied volailiy. Saisical crieria rejec he hypohesis ha implied volailiy is informaionally efficien wih respec o economeric forecass. Bu dela hedging exercises indicae ha such economeric forecass have no incremenal economic value. hus saisical measures of bias and informaion efficiency are misleading measures of he informaion conen of opion prices. Keywords: gold fuures opion implied volailiy GARCH long-memory ARIMA high frequency JEL subjec numbers: F3 G5 * Research Officer Research Deparmen Federal Reserve Bank of S. Louis P.O. Box 44 S. Louis MO 6366 (34) (34) (fax) neely@sls.frb.org Charles Hokayem John Zhu and Joshua Ulrich provided research assisance. he views expressed are hose of he auhor and do no necessarily reflec official posiions of he Federal Reserve Bank of S. Louis or he Federal Reserve Sysem. Any errors are my own.

3 Implied Volailiy from Opions on Gold Fuures: Do Economeric forecass Add Value or Simply Pain he Lilly? Absrac: Consisen wih findings in oher markes implied volailiy is a biased predicor of he realized volailiy of gold fuures. No exising explanaion including a price of volailiy risk can compleely explain he bias bu much of his apparen bias can be explained by persisence and esimaion error in implied volailiy. Saisical crieria rejec he hypohesis ha implied volailiy is informaionally efficien wih respec o economeric forecass. Bu dela hedging exercises indicae ha such economeric forecass have no incremenal economic value. hus saisical measures of bias and informaion efficiency are misleading measures of he informaion conen of opion prices.

4 Gold has capured he imaginaion for housands of years. Ye despie he growing imporance of derivaives relaively lile research has been done on he gold opions marke. Beckers (984) Ball orous and schoegl (985) and Followill and Helms (990) sudied wheher gold opions prices obeyed boundary and pariy condiions using relaively shor samples of prices from European Opions Exchange and/or COMEX. Cai Cheung and Wong (00) looked a he inraday reacions of gold prices o news. Bu here has been lile research on he informaion conen of gold opions prices alhough Szakmary Ors Kim and Davidson (003) include gold in a broad sudy of he informaion conen of many commodiies. his paper seeks o fill his gap in he lieraure wih a comprehensive sudy of implied volailiy (IV) from opions on gold fuures from he COMEX division of he New York Mercanile Exchange. Opion prices depend on he expeced volailiy of he underlying asse reurn. Laane and Rendleman (976) showed ha an opions pricing model can be invered o provide he volailiy of he underlying asse unil expiry called implied volailiy (IV). Laer papers showed ha under risk-neural pricing IV should be approximaely he condiional expecaion of he realized volailiy (RV) unil expiry of he underlying asse. Alhough IV is no a raded asse researchers use his relaion o moivae he sudy of IV wih a very loose appeal o he efficien markes hypohesis (EMH): If IV were no an unbiased and informaionally efficien forecas of realized volailiy one could generae excess reurns by hedging or rading wih beer forecass. Wih his moivaion many auhors have invesigaed he predicive properies of IV in a variey of markes. Mos such research has concluded ha IV is a good bu biased forecas of realized volailiy (RV) unil expiry in hose markes in which opion wrier can hedge easily like he COMEX marke for opions on gold fuures. Evidence on informaional efficiency has Davidson Kim Ors and Szakmary (00) sudy he proper ime scale for opions on he same asses.

5 been mixed. he consisen finding ha IV is a biased forecas of RV has proved puzzling. Of course ess of he properies of IV are implicily join ess of marke efficiency and he esing procedures. IV s apparen bias and informaional inefficiency have promped much research on he properies of opion pricing models and he economeric echniques. his paper exends ha research by applying recen advances in volailiy/opions research o examine why IV is biased in gold markes and wheher is bias and informaional inefficiency maers economically. Heson s (993) sochasic volailiy (SV) opions pricing model provides he expecaion of RV. High-frequency price daa precisely characerize volailiy. More sophisicaed economeric models es he informaional efficiency of IV. None of he hypoheses considered for he bias of IV is a plausible explanaion. Specifically errors-in-variables sample selecion bias poor properies of he es saisics and a price of volailiy risk model fail o explain he bias and informaional inefficiency of IV. Bu simulaions show ha persisence in he IV process migh plausibly generae much of he bias. One se of ess horizon-by-horizon ess of informaional efficiency ofen fail o rejec he null as suggesed by Chrisensen and Prabhala (998). his paper argues however ha such failures do no indicae ha horizon-by-horizon ess have beer small sample properies. Insead horizon-by-horizon ess lack power o rejec any hypohesis of ineres. More fundamenally dela hedging exercises supplemen he saisical crieria o assess he economic value of alernaive volailiy forecass. Economeric forecass do no improve dela hedging performance despie he fac ha IV fails o subsume hose forecass by saisical crieria. he conradicory inference from saisical and economic crieria underscores he imporance of assessing he informaion conen of IV wih he more relevan measure.

6 . Opion prices and realized volailiy Implici variance from he Black-Scholes formula he Black-Scholes (97) opion pricing formula which counerfacually assumes consan volailiy underlies almos all research on he forecasing properies of IV. Hull and Whie (987) provide he jusificaion for using a consan-volailiy model o predic SV: If volailiy evolves independenly of he underlying asse price and no priced risk is associaed wih he opion he correc price of a European opion equals he expecaion of he Black-Scholes (BS) formula evaluaing he variance argumen a average variance unil expiry: BS BS () C ( S V ) C ( V ) h( V ) dv = E[ C ( V ) V ] = σ where he average volailiy unil expiry is denoed as: V = Vτ dτ. 3 Baes (996) approximaes he relaion beween he BS IV and expeced variance unil expiry wih a aylor series expansion of he BS price for an a-he-money opion (see Appendix A): () Var( V ) ( E V ) ˆ σ BS E V. 8 ha is he BS IV ( σ ) undersaes he expeced variance of he asse unil expiry ( E ). BS his bias is very small however. Noe ha () depends on () which assumes ha volailiy risk is unpriced; BS IV is more properly called risk-neural IV. V Equaion () implies ha he BS IV approximaes he condiional expecaion of RV ( V ). his implies ha BS IV is an unbiased esimaor of RV ha {α β } = {0 } in he following: Garcia Ghysels and Renaul (003) survey recen opions pricing models. Whaley (003) looks a he wider derivaives lieraure. 3 Romano and ouzi (997) exend he Hull and Whie (987) resul o include models ha permi arbirary correlaion beween reurns and volailiy like he Heson (993) model. Because reurns and volailiy on gold fuures have very low correlaion he Romano and ouzi (997) adjused 3

7 (3) σ = α + β σ + ε σ RV RV IV where is RV from o and is IV a for an opion expiring a. σ IV he second hypohesis is also moivaed by (). If IV is he condiional expecaion of RV hen IV is an informaionally efficien forecas of RV. Researchers ofen invesigae his wih he following encompassing regression: (4) σ = α + β σ + β σ + ε σ RV RV IV FV where is RV from o he expiraion of he opion a is he IV from o and σ is some alernaive forecas of variance from o. he coefficien esimaes ( ˆ βˆ ) FV σ IV measure he incremenal forecasing value of he IV and economeric forecas. A non-zero esimae of β rejecs he null ha IV is informaionally efficien wih respec o ha forecas. 4 β he Properies of implici volailiy Researchers esimaing versions of (3) have found ha αˆ is posiive and βˆ is less han one for many asse classes and sample periods (Canina and Figlewski (993) Lamoureux and Lasrapes (993) Jorion (995) Fleming (998) Chrisensen and Prabhala (998) Szakmary Ors Kim and Davidson (003) Neely (004)). ha is IV is a biased and overly volaile predicor of RV: A given change in IV is associaed wih a larger change in he RV. ess of informaional efficiency (4) provide more mixed resuls. Kroner Kneafsey and Claessens (993) concluded ha combining ime series informaion wih IV could produce beer forecass han eiher echnique singly. Blair Poon and aylor (00) discover ha hisorical formula is exremely close o he BS formula for he relaively shor-erm opions in his paper. 4 Esimaing (3) wih he sandard deviaion of asse reurns and he implici sandard deviaion (ISD) raher han variances provides similar inference o regressions done wih variances. While previous versions of his paper used ISDs he curren version uses variances for consisency wih he price of variance risk model in Chernov (00). 4

8 volailiy provides no incremenal informaion o forecass from VIX IVs. Li (00) and Marens and Zein (00) find ha inraday daa and long-memory models can improve on IV forecass of RV in currency markes. Szakmary Ors Kim and Davidson (003) find ha IV in gold markes is efficien wih respec o hisorical volailiy and a GARCH forecas. Finally Neely (004) finds ha IV is no efficien by saisical crieria in foreign exchange markes. Several hypoheses have been pu forward o explain he condiional bias: errors in IV esimaion sample selecion bias esimaion wih overlapping observaions and poor measuremen of RV. Perhaps he mos popular soluion is ha volailiy risk is priced. his heory requires some explanaion. he Price of volailiy risk o illusrae he volailiy risk problem consider ha here are wo sources of uncerainy abou he value of an opion in a SV environmen: he change in he price of he underlying asse and he change in is volailiy. An opion wrier mus ake a posiion boh in he underlying asse (dela hedging) and in anoher opion (vega hedging) o hedge boh sources of risk. If he invesor only hedges wih he underlying asse no vega hedging hen he porfolio reurn depends on volailiy changes. If volailiy flucuaions represen a sysemaic risk hen invesors mus be compensaed for exposure o hem. In his case he Hull-Whie resul () does no apply and he BS IV is no even approximaely he expeced objecive variance as in (). he idea ha volailiy risk migh be priced has been discussed for some ime: Hull and Whie (987) and Heson (993) consider i. Lamoureux and Lasrapes (993) argued ha a price of volailiy risk was likely o be responsible for he bias in IV from opions on socks. he volailiy risk premium argumen ress on he facs ha volailiy is sochasic opions prices depend on volailiy and risk premia are ubiquious in financial markes. If cusomers desire a 5

9 ne long posiion in opions for business hedging and opion wriers hedge heir exposure o volailiy by buying oher opions some agen mus sill hold a ne shor posiion in opions and hey will be exposed o volailiy risk. IV s bias could be due o volailiy risk. On he oher hand here seems lile reason o hink ha volailiy risk iself should be priced. While he volailiy of he marke porfolio is a priced facor in he ineremporal CAPM (Meron (973) Campbell (993)) i is more difficul o see why volailiy risk in commodiy markes should be priced. One mus appeal o limis-of-arbirage argumens (Shleifer and Vishny (997)) o jusify a non-zero price of gold fuures volailiy risk. radiionally empirical work has assumed ha volailiy risk could be hedged or is no priced. Bu recen research has reconsidered he role of volailiy risk in opions and equiy markes (Poeshman (000) Baes (003) Benzoni (00) Chernov (00) Pan (00) Bollerslev and Zhou (003) Ang Hodrick Xing and Zhang (003) and Neely (004)). Poeshman (000) for example direcly esimaed he price of risk funcion from SPX opions daa and hen consruced a measure of IV unil expiry from he esimaed volailiy process. Benzoni (00) finds evidence ha S&P 500 variance risk is priced in is opion marke. Using differen mehods Chernov (00) also marshals evidence o suppor his price of volailiy risk hesis for equiy and foreign exchange. his paper exends his research by examining wheher a non-zero price of volailiy risk can explain he bias in IV for fuures on gold. 3. he Daa his paper uses four kinds of daa o invesigae wheher IVs from gold opions prices are biased and informaionally efficien: daily selemen prices on gold fuures daily opions on gold fuures high-frequency (30-minue) reurns on spo gold prices and daily U.S. ineres raes from he Bank for Inernaional Selemens. he high frequency daa begin on January 987 6

10 and end on December he COMEX division of he New York Mercanile Exchange provided daily daa on gold fuures conracs and opions on hose conracs. hese fuures conracs expire in February April June Augus Ocober and December. he opions conracs expire on he second Friday of he monh before he fuures conrac delivery monh. o consruc a series of he mos liquid conracs he conrac daa are spliced in he usual way a he beginning of each opion expiraion monh. ha is on each rading day of January and February he selemen price colleced a :00 p.m. cenral ime and daily range (high minus low price) for he April fuures conrac are colleced. In addiion he srikes and selemen prices for he wo neares-he-money pus and wo neares-he-money calls on he April fuures conrac are also colleced. he opions on he April fuures conrac expire in March. For each rading day in March and April daa are colleced on June fuures conracs and opions on hose June conracs ha expire in May. his procedure collecs daa on six conracs each year wih five o 53 business days o opion expiry. he usual daily volailiy measure exraced from fuures prices is as follows: (5) F σ RV = ln = r F where F is he appropriae fuures conrac selemen price on dae and r is he squared log reurn on dae. he annualized fuures measure of RV unil expiry is he annualized mean square of he daily reurns: 5 (6) σ RV = r + i. + i= 5 As a check on robusness some exercises have been redone wih opions daa ha begin on Ocober 4 98 and end on July Inraday price daa are no available over he exended period so resuls are shown for a period in which comparable daa are available. 7

11 Olsen and Associaes provided he high-frequency reurns on he spo gold prices. he daily measure of volailiy is he sum of he 48 squared 30-minue reurns over each day from :00 p.m. cenral ime o :00 p.m. cenral ime (Anderson and Bollerslev (998)). 6 he inraday (high frequency) volailiy measure for day can be wrien as: (7) σ RV = r. 48 i= i he high-frequency variance measure unil expiry is consruced in he same way as he daily measure in (6). Andersen and Bollerslev (998) argue ha such high-frequency measures more closely approximae he unobserved volailiy process han does he sandard deviaion of daily reurns. Poeshman (000) for example shows ha such a high frequency measure eliminaes ½ he bias in he predicions of S&P 500 (SPX) index opions. One migh consider inraday volailiy esimaes as a complemen o daily volailiy raher han a subsiue. Wheher one is ineresed in he forecasing performance of IV wih respec o inraday or daily volailiy daa migh depend on he applicaion. 4. Economeric mehodology Consrucing implied variance he Heson (993) SV pricing model provides he benchmark measure of IV under he assumpion ha volailiy risk is unpriced. he SV model posis ha he fuures price and volailiy evolve as follows: (8) df = µ Fd + V Fdϖ S dv = θ κ V d + σ V dϖ (9) ( v v ) v v where F is he fuures price a ; V is he insananeous variance of F s diffusion process dϖ S 6 Resuls wih he 5-minue reurns were similar. 8

12 and d ϖ ν are sandard Brownian moion wih correlaion ρ; and κ v θ v / κ v and v adjusmen speed long-run mean and variaion coefficien of he diffusion volailiy. σ are he he SV opions prices are funcions of ρ κ v θ v σ v as well as asse price (F) srike price (X) ineres raes (i) ime o expiry (-) and insananeous variance (V). Sarwar and Krehbiel (000) describe how o obain values for ρ κ v θ v and σ v from he discree ime series process. able shows he esimaed parameer vecor. aking ρ κ v θ v σ v F X i and - as given insananeous variance (V()) is chosen each day o minimize he sum of he squared percenage differences beween he SV model implied prices and he selemen prices for he wo neares-o-he-money call opions and wo neares-ohe-money pu opions for he appropriae fuures conrac. 7 4 (( i i )/ Pr i ) (0) V() = arg min SV ( V () ) Pr σ i= where Pr i is he observed selemen premium (price) of he ih opion on day and SV i (*) is he appropriae call or pu formula as a funcion of he IV and he parameers in (8) and (9). 8 Opion prices ha violaed he no-arbirage condiions on American opions prices (C F X and P X F) were discarded. In addiion he observaion was discarded if here was no a leas one call and one pu price. For a few cases he quasi-maximum likelihood (QML) esimaion failed o converge and a bisecing grid search was used o find IVs insead. he grid search esimaes appeared consisen wih IVs found hrough QML esimaion. 7 Using he full sample o esimae ρ κ v θ v and σ v poenially inroduces a look-ahead bias ino he IVs. One could insead derive all parameers from opions prices each day bu his mehod is compuaionally very difficul in many cases and impossible for some. Alhough Chernov and Ghysels (000) find ha relying only on opions daa in pricing and hedging he S&P 500 index conrac is bes experimens indicae ha i is unlikely o make much difference. In pracice he IVs were no very sensiive o he choice of parameers. 8 Bakshi Cao and Chen (997) and Sarwar and Krehbiel (000) apply versions of he sochasic 9

13 Poeshman (000) and Chernov (00) show ha expeced variance unil expiry can be calculaed by applying Io s lemma o e κ V and using he variance process (9). () E ( V ) = E ( Vu ) θ θ κ v ( ) [ e ] v v du = + V κ v κ v ( ) κ v he expeced variance unil expiry in () is he IV used o predic realized variance unil expiry. his procedure eliminaes he biases in (). Baes (996) repors ha using a-he-money opions has become increasingly popular. here are hree reasons for his pracice: ) A-he-money opions prices are mos sensiive o changes in IV meaning ha changes in IV should be refleced in hose opions. ) A-he-money opions are usually he mos liquid. 3) Research has found ha IV from a-he-money opions provides he bes esimaes of fuure realized volailiy (e.g. Beckers (98)). Despie he fac ha researchers have varied he number of opions he ype of opions and he weighing procedure i has been common o rely heavily on a-he-money opions. herefore choosing he wo neares calls and wo neares pus for esimaing IV each day seems o be a reasonable procedure. Alernaive forecass Four ypes of models provide alernaive forecass of RV o es IV s informaional efficiency: auoregressive inegraed moving average (ARIMA) models long-memory ARIMA (LM-ARIMA) models generalized auoregressive condiional heeroskedasic (GARCH) models and ordinary leas squares (OLS) models wih several independen variables. 9 he Bayesian Informaion Crierion (BIC) chose he specific srucure (e.g. he AR and MA orders of he ARIMA model) of each of he four classes of models during an in-sample period 987 volailiy (SV) opion pricing model o hedging and pricing problems. 9 Poon and Granger (003) review he lieraure on forecasing volailiy in financial markes. Pong Shackleon aylor and Xu (003) compare he forecasing abiliy of ARFIMA models wih IV by 0

14 hrough 99 (Schwarz (978)). he in-sample srucure and in-sample coefficien esimaes were hen fixed and used o forecas RV unil expiry in he ou-of-sample period Appendix B describes hese forecass in deail. Summary Saisics able displays he summary saisics for he annualized one-period volailiy of gold prices and is forecass in he lef-hand panel and he analogous saisics for volailiy unil opion expiraion in he righ-hand panel. While he empirical forecasing work will measure volailiy wih variance o be consisen wih Chernov s (00) price of risk model he summary saisics in able presen he more easily inerpreable sandard deviaion measures. he op panel measures daily volailiy wih he annualized roo of daily sums of log 30-minue squared changes while he boom panel uses he annualized roo sum of squared changes in daily log fuures prices. All saisics are annualized and in percenage erms. RV Mean one-sep realized volailiy (σ ) is 0.44 percen per annum by he highfrequency measure and 8.40 percen per annum by he daily fuures price measure. he mean of he one-sep-ahead forecass are higher han acual volailiy bu he forecas means are based only on in-sample daa no he whole sample as he saisics in able. he forecass are of course less variable han he realized volailiy. he one-sep-ahead forecass in he lef-hand panel have firs-order auocorrelaion coefficiens (row AC) ha range from 0.6 o Comparing he op panel o he boom panel shows ha he daily high frequency volailiy RV measure (column labeled σ ) is also much more highly auocorrelaed han he measure consruced from daily fuures prices. he former has firs-order auocorrelaion of 0.55 o he mean-squared error and R merics.

15 laer s figure of 0.. his is consisen wih Andersen and Bollerslev (998) who argue ha high frequency volailiy more precisely measures unobserved volailiy han daily volailiy. Figure illusraes he mean-revering ime series behavior of boh measures of realized HF Fu IV volailiy unil he nex opion expiraion (σ and σ ) and IV (σ ). IV appears o rack boh realized volailiy series reasonably well as one migh expec. 5. esing for bias and inefficiency using overlapping observaions Is implici variance an unbiased forecas of realized variance? If IV is he marke s predicion of RV and expecaions are raional hen IV should be an unbiased esimaor of fuure volailiy. ha is {α β } = {0 } in he following model: (3) σ RV = α + βσ IV + ε his paper iniially follows mos previous research in esimaing (3) wih OLS and elescoping samples. 0 For overlapping horizons he residuals in (3) will be auocorrelaed and while OLS esimaes are sill consisen he auocorrelaion mus be deal wih in consrucing sandard errors. Such daa ses are described as elescoping because correlaion beween adjacen errors declines linearly and hen jumps up a he poin a which conracs are spliced. o consruc correc measures of parameer uncerainy his paper follows Jorion (995) in using he following covariance esimaor: ( ) ( ) () ˆ ˆ Σ = X ' X Ω X ' X where Ω ˆ = = ˆ ε X ' X + I( s ) ˆ ε ˆ ε ( X ' X + X ' X ) X is he by K marix of s= = s s s s 0 Chrisensen and Prabhala (998) also esimae versions of (3)and (4) wih feasible generalized leas squares (FGLS) for one shor subperiod bu find i does no help IVs bias or efficiency. See able 6 in ha paper.

16 regressors X is he h row of X εˆ is he residual a ime and I(s) is an indicaor variable ha akes he value if he forecas from period s overlaps wih he forecas from period. able 3 shows esimaes of he coefficiens in (3) for gold opions-on-fuures and realized volailiy in gold marke from 987 o 998 wih robus sandard errors as in equaion (). Consisen wih previous research in oher markes e.g. Jorion (995); Canina and Figlewski (993); Lamoureux and Lasrapes (993); Fleming (998); Chrisensen and Prabhala (998) and Neely (004) he βˆ coefficien is always posiive bu also much less han he hypohesized value of one under he null ha he IV is unbiased. βˆ equals 0.48 and 0.56 for he inraday and daily fuures variance measures respecively. When βˆ is less han one IV is said o be an excessively volaile predicor of subsequen realized volailiy because a given change in IV is associaed wih a smaller change in fuure realized volailiy. he Wald p-values in he sixh column of able 3 consruced wih robus covariance marices srongly rejec ha {αβ} equals {0}. In oher words he bias is saisically significan for boh daa ses. High frequency RV is more closely correlaed wih IV han is daily fuures RV. he R for he inraday daa (op row) is 0.49 while ha for he daily fuures daa is only his is consisen wih he idea ha high frequency volailiy is a less noisy measure of he unobserved underlying volailiy in he marke. Is implied volailiy informaionally efficien? If IV is approximaely he condiional expecaion of RV as implied by () hen i subsumes all publicly available informaion. o es his proposiion one can forecas volailiy using he economeric models described in Appendix B ARIMA LM-ARIMA GARCH OLS o see if any of hese forecass of volailiy over he life of he opion add informaion o IV. ha is one can regress RV on IV and a forecas of variance: 3

17 (4) σ RV = α + βσ IV + β σ FV + ε and es if β is posiive. ha β equals zero (Fair and Shiller (990)). If IV subsumes he oher forecas of RV hen one should fail o rejec able 4 presens he resuls of OLS esimaion of equaion (4) wih elescoping samples and robus sandard errors (equaion ()) over he ou-of-sample period 99 hrough 998. Using only ou-of-sample daa creaes a genuinely ex ane exercise wih which o es he four economeric forecass. From lef o righ he four panels of able 4 display he resuls from forecass wih an ARIMA model a long-memory ARIMA model a GARCH() model and he OLS model. he op row of able 4 measures volailiy wih he daily sums of 30-minue squared reurns o while he boom row uses he daily fuures price variance. Wih he high-frequency measure of volailiy (op row of able 4) he coefficiens on he forecass ( ˆ β ) are always posiive and saisically significan a he five percen level in hree of four cases. Similarly he boom row of able 4 shows ha wih he fuures price as he volailiy measure he coefficiens on he forecass ( ˆ β ) are posiive and saisically significan for wo of he four forecass. he coefficiens on he LM-ARIMA and OLS forecass are significan for boh he inraday and daily realized variance daa. he posiive and saisically significan ˆ β rejec he hypohesis ha IV is informaionally efficien. As wih he bias regressions in able 3 he R s for he high-frequency regressions (op row of able 4) are larger han he corresponding R s for he fuures-volailiy regressions (boom row) in each case. IV is more closely correlaed wih highfrequency volailiy han wih he daily volailiy measure. I is no necessary o make he economeric forecas orhogonal o IV before using i in (4). he saisic on βˆ provides he same inference (asympoically) as he appropriaely consruced F es for he hypohesis ha β = 0. And he F es which is based on he R of he regression is invarian o orhogonalizaion of he regressors. 4

18 6. Why is implici variance biased and informaionally inefficien? IV for gold fuures has been shown o be apparenly biased and inefficien. here are wo ways o explain such a resul: ) failure of he EMH; ) or failure of he esing procedures. Because he EMH seems heoreically difficul o assail a leas wihou appeals o informaion problems economiss have focused heir aenion on he esing procedures including he possibiliy ha a non-zero price of volailiy risk could produce bias and inefficiency. Problems wih he esing procedures fall ino several caegories: ) peso/finance miniser problems; ) measuremen error in IV; 3) sample selecion bias; 4) use of overlapping samples; or 5) a non-zero price of volailiy risk. his secion considers wheher hese issues could plausibly generae he bias and inefficiency. Peso problems Unusual sampling variaion called peso or finance-miniser problems migh generae apparenly biased predicions of realized volailiy. ha is agens migh have raionally priced opions while aking ino accoun exreme bu low probabiliy evens ha were no observed in he sample. Conversely oher low probabiliy evens migh have been observed oo ofen in he sample. For example he marke migh have raionally priced in a low probabiliy of periods of exremely volailiy ha never occurred. If such expecaions increased wih realized volailiy IV would appear uncondiionally and condiionally biased producing overly volaile predicions. I seems unlikely ha sample-specific variaion is o blame for he bias observed in IV because IV s bias is a ubiquious resul across asses and sample periods (Poeshman (000)). Furher he only way o correc for such problems is hrough longer spans of daa; he -year daa se used here is already very long by he sandards of he opions lieraure. 5

19 Measuremen error Measuremen error in IV is an obvious candidae explanaion for is apparen bias and inefficiency. I is well known ha error in he independen variable creaes aenuaion bias; he esimaed regression coefficien is inconsisen smaller in absolue value han he rue coefficien. Chrisensen and Prabhala (998) illusrae ha errors-in-variables could also explain IV s failure o subsume oher forecass. Specifically if boh IV and economeric forecass consiue noisy predicions of RV hen an opimal predicor will pu some weigh on each. he convenional wisdom however is ha here is no much error in IV esimaion (Baes (996)). One source of error is specificaion error from using he wrong opions model. he second source is idiosyncraic error from microsrucure effecs like asynchronous prices and bid-ask spreads in boh he opions and he underlying fuures. Baes (000) decomposes he wo ypes of errors for opions on S&P 500 fuures conracs and concludes ha IV is no very sensiive o he choice of wo-facor pricing models. he IV used in his paper is also no very sensiive o he choice of (risk-neural) opion pricing model. able 5 illusraes he similar summary saisics of he IVs from he hree opion pricing models: he SV pricing model of Heson (993) he Barone-Adesi and Whaley (987) early exercise correcion o he Black (976) model and he Black (976) model. he IVs from he hree models are exremely highly correlaed and have very similar summary saisics. he resuls in his paper are robus o he choice of (risk-neural) opion pricing model. One can measure he error in IV esimaion by examining he maximal difference beween he IVs implied by he wo closes calls and wo closes pus (six possible pairs of IVs) each day. able 6 shows ha hese differences are small. he median difference is abou 0 basis poins h and he 90 percenile is abou 77 basis poins. 6

20 Because hese opions have slighly differen degrees of moneyness hey migh imply differen volailiy (Hull (00)). o remove variaion caused by differen degrees of moneyness one can examine he absolue difference beween IVs from pu-call pairs of opions wih exacly he same srike price on he same day. In he absence of bid-ask spreads ransacions coss or early exercise hese differences should be exacly zero. Indeed hey are very small. he median difference for example is only basis poins and he 90 h percenile is basis poins. Experimens conduced in simulaions no repored for breviy indicae ha error less han percen has almos no effec on he esimaes of bias and efficiency. Sample selecion bias Engle and Rosenberg (000) sugges ha sample selecion bias is responsible for bias in S&P 500 index opions. ha is if one canno observe IV or RV during periods of exreme RV perhaps because liquidiy dries up because of uncerainy hen here will be sample selecion bias in he regression of RV on IV. Selecion bias migh resul if volailiy unil expiry were sysemaically higher or lower on days wih missing IV han on oher days. Bu his is no a problem in he presen daa se. IV is available in all periods for which here are fuures prices. Overlapping samples Chrisensen Hansen and Prabhala (00) argue ha he usual regressions conduced wih overlapping forecass migh produce very poor small-sample esimaes. o invesigae he consequences of such overlapping forecass one can eiher simulae he disribuion of he es saisics under he null hypohesis of unbiased forecass or one can independenly esimae he predicive equaion (3) for each forecas horizon. he simulaion mehod has he advanage of Experimens sugges ha correcing IV esimaes for he volailiy smile made very lile difference in he bias or informaional efficiency of IV. Error generaed by he volailiy smile seems unimporan for he issues of bias and informaional efficiency. 7

21 greaer power in pooling all horizons ogeher. he fixed horizon mehod is compuaionally simpler and does no require one o assume ha he regression has he same coefficien vecor a each horizon. his paper confrons he overlapping observaions problem in boh ways. Wha effecs do auocorrelaion and errors-in-variables have on he IV coefficien? Mankiw and Shapiro (986) and Sambaugh (986) argue ha auocorrelaion and measuremen error in he dependen variable will end o bias he coefficien oward zero. o invesigae hese effecs his paper judges he significance of he parameers by using a plausible daa generaing process o simulae he disribuion of he parameer esimaes under he null as in Mark (995) Jorion (995) Kilian (999) and Berkowiz and Giorgianni (00). Boh GARCH and log-arima models are used o simulae and predic RV and IV unil expiry. he ARIMA model was esimaed and simulaed on a modified logarihmic ransformaion of he daily variance daa because hey are runcaed a zero highly skewed and kuroic. he ARIMA forecass were hen ransformed wih a aylor series expansion o produce approximaely condiionally unbiased forecass of RV unil expiry. Appendix C describes hese ransformaions in deail. he simulaion procedure for he GARCH/ARIMA models were as follows:. Esimae he GARCH/ARIMA model wih he whole sample saving he esimaed coefficiens and residuals.. Consruc 000 simulaed variance samples by boosrapping. 3. For each of he 000 samples consruc RV as he annualized sum unil expiry of he squared reurns. he sample sizes will be he same as hose in able 3. Consruc IV as he opimal muliperiod forecas of RV over he appropriae horizon. 4. Regress simulaed RV-unil-expiry on simulaed IV saving he es saisics. 8

22 5. Examine wheher he coefficiens and es saisics from he real daa are consisen wih he disribuion of he coefficiens and es saisics from he simulaed daa. he simulaed daa were checked o ensure ha he summary saisics especially he auocorrelaions of he simulaed daa were reasonably close o he analogous saisics in he real daa and he simulaed IV was an approximaely unbiased predicor of simulaed RV. able 7 displays he resuls of he Mone Carlo experimen simulaing he regression of RV on IV. he upper panel shows he GARCH- generaing process resuls; he lower panel shows log-arima resuls. he firs four columns summarize he disribuion of αˆ ; columns five o eigh show saisics on he disribuion of ˆ β ; and he final four columns display he percenage of rejecions from he simulaed Wald saisics and he simulaed R s. he simulaed GARCH model generaes considerable bias in he esimaes of β he 5 h perceniles of he ˆ β disribuions are 0.5 o And he Wald saisics rejec he null 7 and 50 percen of he ime. he ˆ β esimaes from he real daa see able 3 are in he lef-hand ail of he simulaed disribuions 97 and 87 percen of he simulaed ˆ β are greaer han hose from he real daa for he inraday and fuures variance measures respecively. he GARCH model does no replicae he bias found in IV. he lower half of able 7 shows ha daa produced under he null of unbiasedness from he log-arima model produces subsanial bias in he esimaes of β bu no as much as he GARCH model. he 5 h perceniles of he ˆ β disribuions are 0.60 o 0.6 for inraday and daily measures. he median esimaes of ˆ β are much larger however ranging from 0.83 o 0.89 and he Wald saisics rejec 4 and percen of he ime. he hree of he four simulaed R disribuions appear o be consisen wih he R s from he acual daa (see able 3). 9

23 Persisen-regressor bias in he GARCH and log-arima daa generaing processes canno explain he whole condiional bias observed in IV. A generaing process wih greaer persisence in volailiy migh have produced he apparenly biased and informaionally inefficien coefficiens ha we observe in he daa. For example he fracionally coinegraed relaion beween IV and RV found by Bandi and Perron (003) migh produce enough persisence. Is IV Unbiased in Horizon-by-Horizon Esimaion? Chrisensen Hansen and Prabhala (00) advocae he second mehod of correcing problems wih overlapping samples: Use nonoverlapping samples fixed forecas horizons. 3 o examine how IV migh vary as a predicor across forecas horizons one can esimae (3) separaely for each horizon (k = ) (3) σ RV = α k + β kσ IV + ε where α and β denoe he coefficiens for a horizon of k days unil expiry. Cases (horizons) k k wih fewer han 0 observaions rae were no esimaed leaving a minimum forecas horizon of seven business days and a maximum horizon of 50 business days. here were 6 o 7 observaions for each forecas horizon. Mos horizons had 7 observaions. Figure shows he series of values for αˆ k ˆk β and he p-values for he likelihood raio (LR) es ha {α k β k} = {0 }. As in he overlapping horizon resuls in able 4 he αˆ k are posiive and he ˆ β k are much less han one. he ˆ β k for he fuures daa (solid line) are usually greaer han hose for he high-frequency volailiy measure (dashed line). he LR ess almos always rejec he null ha {αk β k } = {0 } for boh daa ses. he mean of he horizon-byhorizon ˆk β s in Figure are jus slighly larger and 0.58 han hose in he overlapping resuls in able 4. Conrary o resuls in Chrisensen and Prabhala (998) and 3 here is a modes amoun of overlap a horizons greaer han 50 business days. 0

24 Chrisensen Hansen and Prabhala (00) horizon-by-horizon does no make IV appear significanly less biased. Is IV informaionally efficien in fixed horizon ess? ess of informaional efficiency also suffer from poenially poor properies of overlapping daa ses. One can esimae equaion (4) for fixed horizons o alleviae such problems. (4) σ RV = α k + β kσ IV + β kσ FV + ε. he forecasing model srucure and coefficiens were fixed by a search over he in-sample (987-99) period and only ou-of-sample forecass (99-998) were used o esimae (4). Afer deleing forecas horizons wih fewer han 0 observaions horizons ranged from 8 o 49 business days and had beween 3 and 4 observaions during he ou-of-sample period. he op row of Figure 3 displays he series of coefficiens from maximum likelihood esimaion of he pooled model described by (4) on high-frequency (inraday) volailiy daa. he β coefficiens are very volaile bu posiive a mos horizons. he boom row of Figure 3 ˆk shows he corresponding LR es p-values for he hypohesis ha he β k coefficiens are equal o zero. he LR p-values are someimes less han 0.05 rejecing he null of informaional efficiency for horizons less han 30 business days. he forecass have a fair proporion of rejecions a hese shor horizons. he frequency wih which he ess rejec he null does no seem o clearly answer he quesion as o wheher IV is informaionally efficien for highfrequency volailiy. However wih such a small sample size and correspondingly low power one migh expec low power o rejec he null from such a es. herefore obaining rejecions in one quarer of he cases cass some doub on he null of informaional efficiency. Figure 4 shows he corresponding saisics using daily fuures prices o measure volailiy. he ˆk β coefficiens are exremely volaile and only someimes rejec he null of

25 informaional efficiency. Again his is probably due o he pauciy of observaions involved in esimaing he models and he difficuly of fiing forecasing models o he highly skewed and kuroic squared daily reurn series. he failure o rejec informaional efficiency wih horizonby-horizon ess probably reflecs low power raher han beer small sample properies. A Non-zero price of volailiy risk Risk-neural valuaion requires ha he risks associaed wih a posiion in an opion eiher can be hedged or are no sysemaic. his is no necessarily he case. And i is only under riskneural valuaion ha IV is approximaely he condiional expecaion of RV (). Recen research has considered he possibiliy ha volailiy risk is responsible for he bias found in BS IVs (e.g. Poeshman (000) Benzoni (00) Chernov (00) Neely (004)). Chernov (00) derives an expression for expeced realized variance unil expiry in erms of he implied risk-neural expeced variance unil expiry (see Appendix D). ha is one esimaes he following predicion equaion o see if he coefficien β is equal o one: σ RV = α θ κ θ κ + β σ + γ θ κ θ κ V + ε Q Q M M Q Q M M (5) ( v v v v ) IV ( v v v v ) where he coefficiens α and γ are funcions of ime o expiry and he parameers of he Q Q risk-n eural ( κ ) M M θ and objecive SV processes ( κ ) v v θ. If volailiy risk is no priced he risk-neural and objecive SV processes are he same and he coefficien vecor {α β γ } in (5) equals {00}. In his case he convenional predicion equaion (3) is appropriae. Ciing equaion (5) Chernov (00) argues ha he convenional regression o f averag e RV on he risk-neural IV is misspecified because insananeous variance which is heavily correlaed wih he risk-neural IV is omied. his omission biases he coefficien (β ) on riskneural IV downwards. here are poenially a leas ways o esimae equaion (5). he firs mehod is o v v

26 Q Q esimae he hree hyperparameers ( θ κ β ) v v condiional on he ime o expiry ( ) and he parameers from he ime series esimaion of he process ( κ M θ M ) in (5) o help consruc he coefficiens. his imposes he model s resricions across horizons requires one o esimae only 3 free parameers hen es wheher he daa rejec ha β k is equal o one. σ = α κ θ + β k σ + δ κ θ V + ε Q Q Q Q (6) R k ( ) IV k( ) k his mehod is poenially powerful geing precise esimaes bu imposes resricions on he funcional form ruling ou jumps in volailiy for example. he second mehod of esimaing he price-of-volailiy-risk model is o esimae he parameer vecor {α β γ } for each horizon wih no cross-horizon resricions. One can es wheher he daa rejec he model s implicaion ha risk-neural IV is unbiased afer he inclusion of he volailiy risk premium. ha is one ess if β =. Eiher esimaion mehod requires esimaes of insananeous variance as well as risk-neural IV. Insananeous variance is aken o be he daily volailiy esimae from inraday daa. Because his measure and o a lesser exen he IV regressors are esimaed wih error one sho uld use insrumenal variables o esimae (6) (Chernov (00)). he BIC selecs insrumens from lagged values of he esimaed insananeous variance IV and forecass of RV. able 8 presens he resuls of Generalized Mehod of Momens (GMM) (Hansen (98)) esimaion of he parameers (6) accouning for he overlapping observaions in he sandard error (equaion ()). -ess clearly rejec he null ha IV is an unbiased forecaser. Esimaing only hree hyperparameers imposes fairly srong resricions on he model however. he price-of-volailiy-risk model migh do beer if {α k β k γ k } were esimaed separaely for each horizon. his is similar o Chernov s procedure excep ha he esimaed only one forecas horizon (one-monh). 3

27 Figure 5 shows he resuls of esimaing (6) by GMM wih insrumenal variables. he model rejecs he null ha ˆk β equals one for all bu a few horizons. Curiously one of he few horizons ha fails o rejec unbiasedness is he one-monh horizon (0- business days) for which Chernov also failed o rejec unbiasedness wih equiies and foreign exchange. Permiing a price of volailiy risk as in ( 6) does no resolve he puzzle of IV s bias for gold fuures. o examine he robusness of he resuls Figure 5 was reconsruced under alernaive assumpions including a range-based and moving-average based esimaes of insananeous variance and OLS esimaion of he model. None of he alernaive esimaion mehods resuls omied for breviy suppor he unbiasedness hypohesis. 6. Economic implicaions of inefficiency: racking error Judging he informaion conen of opions prices wih saisical measures has never been very ighly moivaed. In paricular he frequen rejecion of bias and efficiency does no ell us abou he economic significance of hese shorcomings. A more relevan es of IV s forecasing abiliy is wheher economeric forecass can usefully augmen he IV in dela hedging. o examine his one can compare he dela hedging racking error wih wo measures of variance: ) only IV or ) an ex ane funcion of IV and variance from economeric forecass. 4 he racking error from dela hedging a call opion on fuures is calculaed as follows:. In he firs period of a conrac he agen sells a call opion pus he proceeds ino bonds and akes a posiion of () unis in he fuures conrac. he value of his porfolio is iniially zero because he long bond posiion offses he shor call. 4 Bollen and Whaley (003) also examine vega hedging for S&P 500 index opions. his exercise is no pursued here because he aim is o evaluae he conribuion of economeric forecass o dela hedging. Green and Figlewski (999) examine he risks inheren in pricing and hedging opions and he exen o which using a higher han expeced IV can compensae for hese risks. 4

28 F ( 0) F (0) (7) V ( 0 ) V ( 0 ) + V ( 0 ) = C ( ) - C ) = 0 = B C BS X BS( X. On subsequen business days he agen s bond holdings are augmened by ineres on he bond posiion and gains (losses) on he previous fuures posiion. he agen also adjuss he fuures posiion o he curren dela (). he porfolio s value on day is r n (8) ( ) () ( ) 365 F ( ) F V() V e V ( ) F C( () = + + ) B X X where VB(-) is he value of he bonds on - here were n calendar days since he las business day F() is he change in he value of he fuures price from - o (-) is he fuures posiion a - and C() is he change in he call price from - o. 3. On he las day of he conrac he agen closes he fuures posiion. he racking error is he difference beween he value of bond holdings and he opion s value. he exercise aims o deermine wheher one-sep-ahead ex ane economeric forecass can improve dela hedging over a benchmark model using only he implied insananeous variance. he for ecas-augmened model picks he relaive weigh of insananeous volailiy (λ) versus he economeric forecas (-λ) and a consan from a grid o consruc he insananeous variance esimae for he dela funcion o minimize racking error in he in-sample period (987-99). he augmened model s insananeous variance is given by he following: Aug SV F (9) V ( ) = λ V ( ) + ( λ) V ( ) + k where V SV () is he insananeous variance from he SV model V F () is he one-sep-ahead forecas volailiy from one of he four economeric forecasing models and k is a consan. he benchmark model consrains λ = and chooses k o minimize in-sample dela hedging error. Boh models choose a negaive k o make dela larger and more volaile o hedge beer a he daily frequency. 5

29 o implemen he dela hedging experimen a he beginning of each conrac period he call wih he longes ime coninuously near-he-money was chosen as he call o hedge. When ha call lef he money he posiion was closed ou he racking error calculaed and a new call chosen o hedge. All posiions were closed a he end of splicing periods. able 9 shows he percenage improvemen in he ou-of-sample racking error using he four economeric forecass. he racking error is he sum of he absolue racking errors for each opion conrac followed. Sandard errors are calculaed wih he Newey-Wes procedure. In all cases mos weigh was pu on he insananeous variance and relaively lile on he economeric forecass; λ s ranged from 0.4 o 0.9. Six of he eigh poin esimaes are posiive bu none are saisically significan. Consisen wih he idea ha here is more informaion abou variance in inraday daa weighs on he inraday variance forecass are larger han hose from he daily daa. Economeric forecass do no consisenly reduce dela-hedging racking error. his sheds new ligh on rejecions of bias and recen rejecions of informaional efficiency (Li (00) and Marens and Zein (00)). I highlighs he need o evaluae forecass wih he mos relevan cri eria. In he case of opions saisical crieria are simply no as informaive as dela hedging. 7. Conclusions his paper has exploied high-frequency daa and new economeric echniques including long-memory models o examine why IV is an inefficien and biased predicor of he realized volailiy of gold fuures. None of he explanaions previously suggesed imprecise volailiy esimaion overlapping samples sample selecion bias a price of volailiy risk ec. can plausibly explain his bias and inefficiency. Persisen regressor bias can explain some of he bias in overlapping samples. Horizon-by-horizon esimaion usually canno rejec ha IV is informaionally efficien. One 6

30 should no conclude however ha he beer small sample properies of horizon-by-horizon esimaion procedures rescue he unbiasedness hypohesis. Raher he failure o rejec almos cerainly reflecs a lack of power. here are only 3 o 4 observaions per horizon in his sample. While saisical crieria judge IV o be a biased and inefficien predicor of realized variance implied insananeous variance is informaionally efficien by an economic crierion dela hedging racking error. his underscores he poin ha choosing he proper crierion is crucial in judging he informaion conen of IV. Using economeric forecass o supplemen IV in dela hedging jus gilds he lily. 7