Relationship between Implied and Realized Volatility of S&P CNX Nifty Index in India. Siba Prasada Panda 1. Niranjan Swain 2. D.K.

Transcription

1 Relaionship beween Implied and Realized Volailiy of S&P CNX Nify Index in India Siba Prasada Panda 1 Niranjan Swain 2 D.K. Malhora 3 Absrac Measures of volailiy implied in opion prices are widely believed o be he bes available volailiy forecass. According o he efficien marke hypohesis, since implied volailiies are calculaed based upon oday s pricing informaion, hey conain he bes informaion abou he marke. Therefore, implied volailiies are considered as he bes represenaion of marke expecaions. Recenly, Chrisensen and Prabhala (1998) found ha implied volailiy in a-he-money one monh OEX call opions on S&P 100 index is an unbiased and efficien forecas of ex-pos realized index volailiy afer he 1987 sock marke crash. In his paper, we examine he informaion conen of call and pu opions on he S&P CNX Nify index. We examine one monh ahe-money call opion from 4-June-2001 o 28-oc We find ha implied volailiy conains more informaion han pas realized volailiy. In oher words he predicabiliy of implied volailiy is more han ha of pas realized volailiy. In fac, implied volailiy remains significan even in he muliple regressions where hisorical volailiy is included. Thus, we find ha i is an efficien albei slighly biased esimaor of realized reurn volailiy. Keywords: Implied Volailiy, Realized Volailiy, Two Sage Leas Square JEL Classificaion: C22, C53, G10 1 Corresponding auhor : TransMarke Group Research (India) Pv. Ld., Mumbai, India, sibg_hcu@yahoo.com 2 BITS, Pilani, Rajashan, India, niranjanswain@bis-pilani.c.in 3 Philadelphia Universiy, School House Lane and Henry Avenue, Philadelphia, MalhoraD@philau.edu 85 Elecronic copy available a: hp://ssrn.com/absrac=

2 Relaionship beween Implied and Realized Volailiy of S&P CNX Nify Index in India 1 - Inroducion The correc valuaion of derivaives is of crucial imporance for praciioners in any financial marke. Volailiy of reurns is a key inpu in he valuaion of opions. For a sophisicaed rader, opions rading is volailiy rading and he rader who has bes volailiy forecas is likely o be he mos successful rader. There are several mehods o predic he fuure volailiy. Some measures of volailiy express he volailiy of a ime series using only realizaions of ha ime series o dae. These volailiy measures are backward-looking in he sense ha hey rely on he hisory of prices. Unlike hese hisorical measures, Forward-looking measures of volailiy rely on curren prices, which incorporae all available informaion abou fuure prices. Alernaively saed, since hese curren prices are deermined by he bes and mos up-o-dae informaion, hey reflec paricipan s expecaions abou fuure marke condiions. Under a raional expecaions assumpion, he marke uses all he informaion available o form is expecaions abou fuure volailiy, and, hence, he marke opion price provides he marke s rue volailiy esimae. Furhermore, if he marke is efficien, he marke s esimaed, implied volailiy is he bes possible forecas given he currenly available informaion, which means ha all informaion necessary o explain fuure realized volailiy generaed by all oher explanaory variables in he marke informaion se should be subsumed in he implied volailiy. The objecive of his sudy is o invesigae he predicive power of implied volailiy agains he pas realized volailiy of S&P CNX Nify 4 index opion in India. We disinguish our sudy from previous work in wo ways. Firsly, we consider volailiy daa, sampled over a longer period (42 monh) of ime, while previous sudies cover a ime 4 The S&P CNX Nify is he leading index for large companies on he Naional Sock Exchange of India. I consiss of 50 companies represening 24 secors of he economy, and represening approximaely 77% of he raded value of all socks on he Naional Sock Exchange of India. 86 Elecronic copy available a: hp://ssrn.com/absrac=

3 period of 3 monhs only. This increases he saisical power and allows for evoluion of efficiency of he marke dealing wih S&P CNX Nify index opions ha were inroduced in Secondly, we sample he implied volailiy and realized volailiy series a near monh (one monh) frequency. This enables consrucion of volailiy series wih non-overlapping daa wih exacly one implied and one realized volailiy ha covers each ime period in he sample. Res of he paper is organized along he following lines. In secion II, we discuss previous sudies. Secion III describes how volailiy series are consruced and provides descripive saisics for hese series. Secion IV describes he mehodology used in his sudy. In Secion V, we presen he empirical resuls, and Secion VI concludes our sudy. 2 - Lieraure Review Early sudies of he informaion conen of ISDs (Implied Sandard Deviaions) show ha implied volailiy conains subsanial informaion for fuure volailiy. Laane and Rendleman (1976), Chiras and Manaser (1978), and Beckers (1981), for example, regress fuure volailiy on he weighed implied volailiy across a broad sample of Chicago Board Opions Exchange (CBOE) socks, and find ha opions conain volailiy forecass ha are more accurae han hisorical measures. These sudies were performed shorly afer he 1973 beginning of he CBOE opion marke and, herefore, use a relaive shor ime span and focus on cross secions raher han ime series predicions. These papers essenially documen ha socks wih higher implied volailiies also have higher ex-pos realized volailiy. Sco and Tucker (1989) repor some predicive abiliy in ISDs (Implied Sandard Deviaions) measured from PHLX currency opions, bu heir mehodology does no allow formal ess of hypohesis. They use OLS (Ordinary Leas Square) regression wih 5 currencies, 3 mauriies, and 13 differen daes. Because of correlaions across observaions, he usual OLS sandard errors are severely biased, hereby invalidaing hypohesis es. Day and Lewis (1992) analyze opions on he S&P 100 index from 1983 o 1989, and find ha he ISD (Implied Sandard Deviaion) 87 Elecronic copy available a: hp://ssrn.com/absrac=

4 conain significan informaion conen for weekly volailiy, alhough no necessarily higher han ha of ime series models. This approach, however, ignores he erm srucure of volailiy since he reurn horizon is no mached wih he life of he opion. Lamoureux and Lasrapes (1993), who examine opions on en socks wih expiraions from 1982 o 1984, conclude ha implied volailiy is biased and inefficien. Boh of hese sudies use overlapping sample and addiionally, are characerized by a mauriy mismach problem. Lamoureux and Lasrapes (1993) examine one-day-ahead and Day and Lewis (1992) examine one-week-ahead predicive power of implied volailiies compued from opions ha have a much longer remaining life (up o 120 rading days in he former and 36 rading days in he laer). Therefore, he resuls are hard o inerpre. Canina and Figlewski (1993) regress he volailiy over he remaining conrac life agains he implied volailiy of S&P 100 index opions over 1983 o They repor ha ISDs (Implied Sandard Deviaions) conain lile predicive power for fuure volailiies appear o be even worse han simple hisorical measures. Empirical research conduced on currency opions, on he oher hand, in general concludes ha he implied volailiy on shor mauriy conracs performs beer in forecasing fuure volailiy and conains informaion ha is no presen in hisorical volailiy. Jorion (1995) invesigaes he informaion conen of implied volailiy from currency opions raded on he Chicago Mercanile Exchange. Jorion (1995) finds ha saisical ime series models are ouperformed by he volailiy implied in shor-erm opions alhough implied volailiy appear o be a biased forecas. Wih he opions raded on he over-he couner marke, Galai and Tsaasaronis (1996) also evaluae he predicive power of volailiy implied in currency opions and show ha implied volailiy of shormauriy opions performs beer in forecasing fuure volailiy, alhough i is a biased esimaor. For longer horizons hey find ha neiher hisorical nor implied volailiy provides a good forecas of fuure volailiy. Chrisensen and Prabhala (1998) also examine he predicive power of implied volailiy on S&P 100 index opions. In conras o previous work, hey find ha implied volailiy ouperforms hisorical 88

5 volailiy in forecasing fuure volailiy. They aribue he difference in heir resuls from hose of Canina and Figlewski o he use of longer ime series and non-overlapping daa. They also provide evidence ha here is regime shif following he oc-1987 sock marke crash, wih implied volailiy being more biased before he crash han afer he crash. Fleming (1998) examines he performance of he S&P 100 implied volailiy as a forecas of fuure sock marke volailiy. The resuls indicae ha he implied volailiy is an upward biased forecas, bu also ha i conains relevan informaion regarding fuure volailiy. Kai (2002) compares he predicive abiliy of implied volailiy and hisorical volailiy for hree major currencies. Kai shows ha hisorical volailiy a horizons ranging from one monh o six monhs conains more informaion regarding fuure realized volailiy. Szakmary, Ors, and Kim (2003) use daa from 35 fuures opions markes from eigh separae exchanges and es for he predicive power of implied volailiies in he underlying fuures marke. They repor ha implied volailiies ouperform hisorical volailiy as a predicor of he realized volailiy for a large majoriy of commodiies included in heir sudy. Gio (2002) assesses he efficiency, informaion conen and unbiasedness of volailiy forecass based on he VIX/VXN implied volailiy indices, RiskMerics and GARCH ype models a he 5-, 10- and 22-day ime horizon. His empirical applicaion focuses on he S&P100 and NASDAQ100 indices. He also deals wih he informaion conen of he compeing volailiy forecass in a marke risk (VaR ype) evaluaion framework. The performance of he models is evaluaed using LR, independence, condiional coverage, and densiy forecas ess. His resuls show ha volailiy forecass based on he VIX/VXN indices have he highes informaion conen, boh in he volailiy forecasing and marke risk assessmen frameworks. Because hey are easy-o-use and compare very favorably wih much more complex economeric models ha use hisorical reurns, he argues ha opions and fuures exchanges should compue implied volailiy indices and make hese available o invesors. Claessen and Minik (2002) examine alernaive sraegies for predicing sock marke volailiy. Employing German DAX-index reurn daa, hey find ha pas reurns do no conain useful informaion 89

6 beyond he volailiy expecaions ha is already refleced in opion prices. Becker, Clemens and Whie (2006) examine wheher a publicly available and commonly used implied volailiy index, he VIX index (as published by he Chicago Board of Opions Exchange) is in fac efficien wih respec o a wide se of condiioning informaion. Resuls indicae ha he VIX index is no efficien wih respec o all he elemens in he informaion se ha may be used o form volailiy forecass Pong, Shackleon, Taylor, and Xu (2004) find significan incremenal informaion in hisorical forecass beyond he implied volailiy informaion o predic fuure realized volailiy for forecass horizons up o one week. Jiang and Tian (2005) use S&P 500 index opions o compare he predicive abiliy of implied volailiy and hisorical volailiy. They conclude ha implied volailiy is a more efficien forecas for he fuure realized volailiy. 3 - Daa and Sampling Procedure 3.1 Daa Descripion We consider S&P CNX Nify index opions which sared rading from June 4, 2001 under Naional Sock Exchange (NSE). The index consiss of 50 highly raded scrips drawn from diverse indusries and markes. The index opions conracs have a maximum of 3 monhs rading cycle (1 s monh-near monh, 2 nd monh-middle monh and, 3 rd monh- Far monh). New conracs are inroduced on he rading day following he expiraion of he near monh conrac. Exchange provides a minimum of seven srike prices for every opion ype (call and pu) during he rading monh. Our empirical analysis focuses on S&P CNX Nify index opions. We consider he daily closing price basis ne of dividend of S&P CNX Nify index and he index call opion closing price wih 17 rading days o expire (Near monh) from 4 h July-2001 o 28 h Oc The S&P CNX Nify has been obained from he CD- ROM provided by NSE and he NSE websie (nseindia.com). 90

7 3.2 Sampling Procedure By convenion, S&P CNX Nify opions expire las Thursday of every monh. The Friday ha immediaely follows he expiraion, we record he S&P CNX Nify index level, S. If Friday is a holiday, hen we consider he nex business day. On he same dae, we locae a call opion ha is expiring nex monh and is closes o being a-he-money. We record he price, C of his call as well as is srike price, K. This opion expires on he las Thursday of he following monh ( +1); he nex ( + 1) call opion is sampled on he Friday ha immediaely follows he expiraion. An enire sequence of opion prices is consruced in his manner. The key feaure of his sampling procedure is o remove he non-overlapping problem. 4 Mehodology 4.1 Forward Looking Measure of Volailiy The implied volailiy of an opion is defined as he expeced fuure volailiy of he underlying asse over he remaining life of he opion ha equaes he fair value of he opion implied by a paricular model o he opion s acual marke price. Many sudies conclude ha measures of opion implied volailiy are, indeed, he bes predicor of fuure volailiy. Unlike ime series measures of volailiy ha are enirely backward-looking, opion implied volailiy is backed-ou of acual opion prices which, in urn, is based on acual ransacions and expecaions of marke paricipans- and, herefore, is inherenly forward-looking. This measuremen incorporaes he mos curren marke informaion and, herefore, i should reflec marke expecaions beer han he hisorical measure. 5 In he Black-Scholes model, he one unobserved parameer is he volailiy of he underlying sock. Theoreically, he proper volailiy inpu in he Black-Scholes model is he insananeous variance of asse reurns, which means he variance of underlying asse s reurn over an infiniesimal ime incremen. Meron (1973) 5 See Fama (1971, 1990) 91

8 shows ha under his inerpreaion, he volailiy implied by he Black- Scholes model can be inerpreed as he expeced fuure insananeous variance of he underlying asse s reurn over he remaining life of he opion. 6 To find ou implied volailiy of European call and pu opion wih a given marke price of he opion, curren sock price and ineres rae, he value of volailiy is derived by subsiuing hose observed values ino he Black-Schole model. The resuling value of σ is called he Black-Scholes implied volailiy 7, because ha number represens he volailiy of he underlying asse ha is implied by quoed opion price and he Black-Scholes model. In his sudy, we examine he relaionship beween volailiy implied by an opion price and subsequenly realized index reurn volailiy. We, herefore, consruc a ime series of realized volailiy. Realized volailiy is calculaed as he sandard deviaion of he daily index reurn during he remaining life of he opion, he period covered by he implied volailiy. Since i is assumed ha spo prices are log normally disribued, reurns have been calculaed according o heir log differences in prices and are, herefore, coninuously compounded. Therefore, he following mehod has been used o calculae daily index reurn. r k = ln S k ln S k 1 (1) T 1 Where, S k denoes he spo price on day k. Le r = r, k denoes he T k = 1 sample mean of he index reurns in monh. The annualized realized sandard deviaion for index reurn for he monh is given by: 1 T 2 ( r, k r ) T k = 1 R = (2) Where, k runs from Friday following he las Thursday in monh o he las Thursday in monh ( + 1). Here, T denoes he number of 6 see Meron(1973) 7 Implied volailiy for boh call and pu opions 92

9 rading days o mauriy of opions wih expiraion in monh ( + 1). While implied volailiy is known a he beginning of period, he realized volailiy R is no known unil he end of period (he expiraion day of opion). Finally, he analysis is based on boh volailiy and log-volailiy series, which we denoe by CI as call implied volailiy, PI as pu implied volailiy, LCI as log-call implied volailiy, LPI as log-pu implied volailiy, R as realized volailiy, and LR as log-realized volailiy (he sudy considers log as naural logarihm) a ime. Table 1 provides descripive saisics for he hree volailiy series-- he implied volailiy for S&P CNX Nify index opions and he realized index reurn volailiy. Table 1 Descripive Saisics for each of he six series used in his sudy Saisics Mean Call- Implied Volailiy (CI ) Pu- Implied Volailiy (PI ) Realized Volailiy (R ) Log- Call- Implied Volailiy (LCI ) Log- Pu- Implied Volailiy (LPI ) Log- Realized Volailiy (LR ) Sandard Deviaion Skewness Kurosis Jarque-Bera On an average, he pu implied volailiy is slighly higher han he realized volailiy and call implied volailiy. This is consisen wih Harvey and Whaley (1991, 1992) who sugges ha i may be due o he fac ha buying index pus is a convenien and relaively inexpensive way o implemen porfolio insurance. This leads o an excess buying pressure on index pus o index calls, and, in urn, resuls in a high pu implied volailiy compared o hose obained from he corresponding calls. The daa also shows an ineresing paern in he sandard 93

10 deviaion of he volailiy series. The implied volailiies are less volaile han realized volailiy. Boh implied and realized volailiy are highly skewed and lepokuric, whereas he disribuion of he log-volailiy series are less skewed. The Jarque-Bera es of normaliy, JB, is significanly smaller for he log-ransformed series, which indicaes ha he log-ransformed series conform beer o normaliy han he raw series. The hree volailiy ime series are ploed in Figure 1. Figure 1 Implied Vs Realized Volailiy 0,7 0,6 0,5 0,4 0,3 0,2 0, Monh call pu realised 5 - Empirical Resuls In his secion, we examine he relaionship beween realized reurn volailiy and volailiy implied by he opion price. In he sudy boh linear and log-linear relaionships have been esimaed, because he logarihm of volailiies conforms bes o normaliy and i also enables us o compare our resuls wih he previous lieraure. 94

11 The predicive abiliy of implied volailiy is evaluaed by regressing realized volailiy (R ) on he call implied volailiy and he pu implied volailiy, respecively. R = α 0 + α CI + ε (3) c Where, R denoes he realized volailiy for period and CI denoes he call implied volailiy and for pu opion analysis PI denoes he pu implied volailiy a he beginning of period. While performing a regression using equaion 3, we also consider LR (Log of Realized Volailiy a ime ), LCI (Log of Call Implied Volailiy a ime ), and LPI (Log of Pu Implied Volailiy a ime ) in equaion 3, insead of R, CI and PI, respecively. Specifically, we es hree hypoheses using equaion 3. Firs, if call implied volailiy conains some informaion abou fuure volailiy hen α c should be nonzero. Second, if call implied volailiy is an unbiased forecas of realized volailiy, hen α o = 0 and α c = 1. Finally, if call implied volailiy is efficien, he residuals ε should be whie noise and uncorrelaed wih any variables in he marke informaion se. The same mehodology is used o evaluae he predicive abiliy of pu implied volailiy. Before performing regression analysis, we es for uni roo in all he series, because if volailiy series possesses a uni roo, regressions as specified above are spurious. Uni roo es is carried ou hrough Dickey-Fuller (DF), Augmened Dickey Fuller (ADF) and Phillips-Perron (PP) ess. The resuls of uni roo es given in he Table 2 sugges ha all he variables are saionary a 1-percen level. 95

12 LEVELS Table 2 Tes of Saionariy for each of he six ime series using Dickey-Fuller, Augmened Dickey-Fuller, and Philip-Perron mehods. Variables Wihou Trend Wih Trend DF ADF PP DF ADF PP R -5.22* -3.42**(1) -5.28*(3) -5.27* -3.43***(1) -5.32*(3) CI -4.00* -2.71***(1) -4.03*(3) -4.21* *(3) PI -3.24** -2.91***(1) -3.21**(3) ** ***(3) LR -4.40* -2.74***(1) -4.41*(3) -4.39* *(3) LCI -4.10* -2.66***(1) -4.16*(3) -4.31* *(3) LPI -3.52** -2.86***(1) -3.50**(3) ** **(3) Noe: * Rejec he null hypohesis of a uni roo wih 99% confidence ** Rejec he null hypohesis of a uni roo wih 95% confidence *** Rejec he null hypohesis of a uni roo wih 90% confidence Figures in he brackes agains ADF saisics are he numbers of lags used o obain whie noise residuals, and hese lags are seleced using AIC. In PP es we used he lag lengh 3. This opimal lag lengh is seleced using he Newey-Wes mehod. R is he realized volailiy, CI as call implied volailiy, PI as pu implied volailiy, LCI as log-call implied volailiy, LPI as log-pu implied volailiy, and LR is he log-realized volailiy. Ordinary Leas Square esimaes of equaion (3) for boh volailiy level series and log-volailiy series are repored in Table 3. 96

13 Nify Index in India Froniers in Finance and Economics Vol.5 No1 April 2008, Table 3 Ordinary Leas Square Esimaion of Realized Volailiy Noe: *1% and **5% level of significance. Here, CI and PI are denoes he Black-Scholes call implied volailiy and pu implied volailiy for a-he-money opions on S&P CNX Nify index, measured a he beginning of monh, and R denoes he ex-pos daily reurn volailiy of he index, over he remaining life of he opion. The regressions repored in he righ side of each panel are of he logarihmic form. Where, LR, LCI, and LPI saisfy LR = log R, LCI = log CI, and LPI = log PI. The resuls repored in he able are based on nonoverlapping monhly volailiy observaions for he 42 monhs from 4-june-2001 o 28-oc Numbers in he parenheses denoe asympoic saisics. OLS Esimaes Dependen Variable: Realized volailiy (R ) OLS Esimaes Dependen Variable: Log Realized volailiy (LR ) Independen Inercep CI PI R -1 R 2 DW Independen Inercep LCI LPI LR -1 R 2 DW Variables Variables Call Implied 0.11* 0.42** Naural Log of Call Implied -1.16* 0.34* Volailiy (CI ) (2.48) (1.71) Volailiy (LCI ) (-3.86) (2.14) Pu Implied Volailiy (PI ) 0.10* (2.08) 0.37** (1.83) Naural Log of Pu Implied Volailiy (LPI ) -1.20* (-4.01) 0.40* (2.06) Pas Realized 0.15* Naural Log of Pas -1.27* 0.30** Volailiy (R -1 ) (4.60) (1.02) Realized Volailiy (LR -1 ) (-4.37) (1.87) Call Implied Volailiy (CI ) and Pas Realized Volailiy (R -1 ) Pu Implied Volailiy (PI ) and Pas Realized Volailiy (R -1 ) 0.10* (2.27) 0.10** (1.89) 0.45 (1.33) 0.48 (1.52) (-0.08) (-.47) Naural Log of Call Implied Volailiy (LCI ) and Naural Log of Pas Realized Volailiy (LR -1 ) Naural Log of Pu Implied Volailiy (LPI ) and Naural Log of Pas Realized Volailiy (LR -1 ) -1.08* (-3.31) -1.13* (-3.50) 0.30 (1.23) 0.27 (1.01) 0.09 (0.41) 0.13 (0.62)

14 Using only one independen variable, we observe ha boh he implied volailiies conain more informaion han pas realized volailiy. The slope coefficiens for boh call implied volailiy and log-call implied volailiy are 0.42 and 0.34 and hese are saisically significan. In he case of pu opion, he esimaed value of slope coefficiens for boh pu implied volailiy and is log ransformaion are 0.37 and 0.40 and are saisically significan. The Durbin-Wason saisics (DW) are no significanly differen from wo for boh he cases indicaing ha he residuals from he regression equaions are no auo-correlaed. For he realized volailiy in boh he cases, value of slopes is oo low and he respecive -saisics are insignifican. Hence, i can be concluded ha implied volailiy (boh call and pu) conains more informaion abou fuure volailiy han he realized volailiy. However, i appears o be a biased forecas of fuure volailiy since slope coefficien is differen from uni and inercep is differen from zero for boh he cases (volailiy and log-volailiy). Subsequenly, we compare he informaion conen in implied volailiy o ha of pas realized volailiy by esimaing he following muliple regression for boh volailiy level and log-volailiy series for call opions and for he pu opions. R = α 0 + α 0CI + α hr 1 + ε (4) OLS esimaes of equaion (4) are repored in Table 3. Pas realized volailiy in isolaion explains fuure volailiy, which is already discussed earlier in his paper. However, once call implied volailiy is added as an explanaory variable, he regression coefficiens for boh pas realized volailiy and pas log-realized volailiy α h drops from 0.16 o 0.01 and from 0.30 o 0.09, respecively. In he case of pu opions, he regression coefficiens for boh pas realized volailiy and pas log-realized volailiy α h drops from 0.30 o 0.13, respecively, bu he -saisic is insignifican in he second case. Neverheless, he slope coefficien for implied volailiy iself remains significan in he muliple regressions and, in paricular, is much more han he coefficien of pas realized volailiy. 98

15 Therefore, our resuls confirm ha S&P CNX Nify index opion implied volailiy is an efficien bu a biased esimaor of realized volailiy. Neverheless, implied volailiy has more predicive power han he pas realized volailiy wheher judged by magniude of he regression slope coefficien or by R 2 for each regression discussed above. From he above findings we can conclude ha he implied volailiy conains informaion abou fuure volailiy beyond wha is conained in he pas realized volailiy. As shown firs by Chrisensen and Prabhala (1998), here is a poenial misspecificaion problem in inerpreing he resuls of OLS encompassing regression presened in Table 3. The misspecificaion is driven by poenial measuremen error implici in implied volailiy. To overcome his difficuly, he presen sudy implemens a wo-sage leas square regression mehod. In he firs sage, implied volailiy on wo insrumenal variables, namely, lagged realized volailiy and lagged implied volailiy are regressed (Equaion (5)). As shown in Equaion (6), we use fied values of implied volailiy as regression in he second sage. In hese cases, for calls and pus, he wo-sage leas square mehod is also applied for heir log ransformaion. Firs sage, CI = α 0 + α cci 1 + α hr 1 + ε (5) We examine his specificaion for a leas wo reasons. Firs, we use specificaion (5) in an insrumenal variable framework o correc error in variable problems in call implied volailiy. Second, we use i o es wheher call implied volailiy is prediced by pas volailiy. If opion prices reflec volailiy informaion, call implied volailiy should no only predic fuure volailiy, bu should also endogenously depend on pas volailiy, since pas and fuure volailiy are posiively relaed. We es his implicaion using regression equaion (5). Second sage, R = β 0 + β CI + ε (6) c 99

16 In he conras o previous sudies [Chrisensen and Prabhala (1998); Chrisensen and Hansen (2002)], he presen sudy does no include he lagged realized volailiy separaely in he second sage. The generaed regressor used in he second regression is a linear combinaion of lagged realized volailiy and lagged call implied volailiy. If we include lagged realized volailiy in a regression along wih he generaed regressor, i would resul in a mulicollineariy problem by consrucion. Similarly, Chrisensen and Hansen (2002) s approach, in which boh call and pu regressors are specified ogeher in he second sage regression, resuls in problem of mulicollineariy. Therefore, our sudy analyzes call and pu implied volailiy separaely. The resuls of he firs sage regression are presened in Table 4, while he second sage resuls are in Table 5. Table 4 Insrumenal Variable Esimaion (IV) mehod Firs sage Regression esimaes for boh call and pu Dependen variable (IC ) Dependen Variable (LIC ) Inercep IC -1 R -1 R 2 DW Inercep LIC -1 LR -1 R 2 DW * (2.66) 0.30* (2.57) 0.35* (4.83) (-1.52) 0.60* (5.60) 0.21** (1.98) Dependen Variable (IP ) Dependen Variable (LIP ) Inercep IP -1 R -1 R 2 DW Inercep LIP -1 LR -1 R 2 DW * (2.09) 0.40* (4.41) 0.50* (7.20) (-0.813) 0.33* (2.99) 0.48* (5.36) Noe: *1% and **5% level of significance. The resuls repored in he able are based on nonoverlapping monhly volailiy observaions for he 42 monhs from 4-June-2001 o 28-Oc Numbers in he brackes denoe asympoic saisics. 100

17 Insrumenal variable for LIC Insrumenal variable for LIP Second sage of Insrumenal Variable Esimaes Analysis for Call opion Dependen variable-r Dependen variable-lr Insrumenal variable for IC IC -1 Insrumenal variable for LIC LIC -1 Inercep IC R 2 DW Inercep LIC R 2 DW (0.4331) 0.80 (1.38) ( ) 0.70** (1.78) Dependen variable-r Dependen variable-lr Insrumenal variable for IC IC -1, R -1 LIC -1, LR -1 Inercep IC R 2 DW Inercep LIC R 2 DW (1.46) 0.51 (1.44) * (-2.18) 0.50* (2.17) Analysis for Pu opion Dependen variable-r Dependen variable-lr Insrumenal variable for IP IP -1 Insrumenal variable for LIP LIP -1 Inercep IP R 2 DW Inercep LIP R 2 DW (1.48) 0.27 (0.7544) ** (-1.87) 0.48 (1.25) Dependen variable-r Dependen variable-lr Insrumenal variable for IP IP -1, R -1 LIP -1, LR -1 Inercep IP R 2 DW Inercep LIP R 2 DW * (2.10) 0.28 (1.15) ** (-2.61) 0.51** (1.99) Noe: *1% and **5% level of significance. The resuls repored in he able are based on nonoverlapping monhly volailiy observaions for he 42 monhs from 4-June-2001 o 28-Oc Number in he brackes denoes asympoic saisics. The firs sage regression shows ha boh he explanaory variables are highly significan in he case of boh call and pu implied volailiy. This signifies ha he pas realized volailiy and pas implied volailiy (boh call and pu) impac implied volailiy. 101

18 For he analysis of second sage regression, we divide Table 5 ino wo pars and each par is again divided ino wo sub pars. Firs par deals wih he call opion and second par deals wih he pu opion and he sub pars deal wih he selecion of insrumenal variables. In he firs sub par we deal wih lagged implied volailiy as an insrumenal variable. In he second sub par we deal wih lagged implied and lagged realized as an insrumenal variables for boh call and pu opions. We use only he lagged implied volailiy (for boh call and pu) as insrumenal variables. If only lagged implied volailiy is allowed as an insrumen, hen all he explanaory power ascribed o implied volailiy in he esimaion is based on informaion backed ou of opion prices. In order o analyze his issue, we repea he insrumenal variable procedure. When consrucing he insrumen, we use fied values from implied volailiy equaion wih lagged realized volailiy excluded. The resuls appear in he firs sub par of Table 5 for he call and pu opion. In he case of call opion, he slope of he call implied volailiies are higher han he OLS regression for he level series, bu hey are no significan; he resuls are high and significan in he log ransformaion case. In he case of pu opion, he pu implied volailiies are lower han he OLS regression for he level series and hey are no significan, bu hey are high and significan in he log ransformaion case. Overall he slope coefficien of call implied volailiy is slighly larger han pu implied volailiy. I also shows ha he slope of he implied volailiy is higher han he OLS esimaion for boh call and pu opion. Thus he resuls show ha implied volailiy conains informaion abou he fuure volailiy bu i is a biased esimaor. 6 Conclusions There are wo major conceps applied for esimaing fuure volailiy one is assessmen from hisorical daa, while he ohers are uilizing opion pricing heory o ge expeced fuure volailiy from opion prices. This paper offers a criical look a he widely held belief ha implied volailiy compued from marke opions prices is an informaionally efficien forecas of he volailiy ha will acually be experienced by he underlying asse from he presen hrough he 102

19 expiraion dae. Based on his background our sudy invesigaes wheher he implied volailiies of S&P CNX Nify call and pu opion predic he fuure realized index reurn volailiy. For his analysis, we consider one monh (near monh) sampling frequency and nonoverlapping daa for 42 monhs, so ha exacly one implied volailiy and realized volailiy esimae perain o each monh. The sudy considers implied volailiies derived from Black-Scholes model of one-monh a-he-money opion on he S&P CNX Nify index. As an exension we separae he analysis of he volailiies implied in pu and call opions. We show ha implied pu volailiy on an average is slighly greaer han implied call volailiy, possibly because of buying pu index opions is a relaively cheaper and convenien way of implemening porfolio insurance. A uni roo es of he call implied volailiy, pu implied volailiy and realized volailiy series suggess ha all he variables are saionary a level. The OLS resuls indicae ha call implied volailiy is a beer forecas han pu implied volailiy. The sudy also uses an insrumenal variable mehod o correc error in variable problems in call implied volailiy and also o es wheher call implied volailiy is prediced by pas volailiy. From boh OLS and insrumenal variables mehods, we find ha pas realized volailiy does no add any informaion beyond wha is already conained in he implied volailiy. The resuls suppor he hypohesis ha he implied volailiy dominaes hisorical volailiy in forecasing realized volailiy, or ha all he informaion conained in hisorical volailiy is being refleced by he implied volailiy, and he hisorical volailiy has no incremenal forecas abiliy. Therefore, we canno rejec he hypohesis ha he volailiy implied by one monh (near monh) a-he-money call opion price is efficien and, albei, slighly biased esimaor of realized reurn volailiy. This biasness may be due o he fac ha he implied volailiy does no conain all he marke informaion. References Black, F., and M. Scholes, 1973, The Pricing of Opion and Corporae Liabiliies, Journal of Poliical Economy, 81,

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