Implied volatility and future market return

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1 Insurance Markes and Companies: Analyses and Acuarial Compuaions, Volume, Issue, 00 Ping Hsiao (USA), Ming Li (USA) Implied olailiy and fuure marke reurn Absrac This sudy examines he predicabiliy of he implied olailiy (IV) of sock opion conracs on he fuure marke reurn. Using reurn and opions daa of he S&P 00 index beween 996 and 008, we find ha when he marke reurn drops significanly, a high IV srongly predics a fuure marke reersal. On he oher hand, when he marke reurn drops only modesly, a high IV acually predics a coninuing marke loss. We, hen, deelop and explore wo rading sraegies based on our findings, which yield much higher risk-adjused reurns han he S&P 500 index. Keywords: implied olailiy, predicabiliy, conrarian rading sraegy, momenum rading sraegy. Inroducion Does he implied olailiy (IV) of sock opions predic sock reurns? The answer o his quesion paricularly perains o a srand of rading sraegy called olailiy iming, and could poenially help inesors o make more accurae asse allocaion in he porfolio. Preious sudies (e.g. Gio, 005; Doran e al., 00) found ha IV was a weak predicor of he fuure marke reurn, alhough many praciioners suspec ha a sronger relaion exiss beween curren IV and fuure reurn. In addiion, i is widely acceped ha olailiy iming can improe porfolio reurns (see Eraker e al., 003; Fleming, Kirby and Osdiek, 999; and Johnnes e al., 00). These sudies hae shown ha curren sae of he condiional olailiy is ery informaie abou fuure daily or weekly reurns. Gien ha IV is a naural measure of he condiional olailiy, here may be a sronger correlaion beween IV and fuure reurns. Howeer, he curren lieraure has mixed resuls regarding his issue. Backus and Gregory (993) repor a decreasing or zero relaion beween fuure marke risk premium and condiional ariance of marke reurn. Whielaw (997) also calibraed reasonable parameers for a negaie relaion in a single facor model. Bu Scruggs (998) shows ha here could be a posiie relaion, if more facors are included in he model. In his paper, we inesigae he relaion beween fuure marke reurn and marke s condiional ariance based on a differen approach. Using a sandard dynamic facor model of reurn proposed by Campbell and Yogo (006), and Fama and French (988), we show analyically ha he sign of he relaionship is nonlinear, i.e., he predicion of IV is no uniersal across all saes of he marke, insead i depends on Ping Hsiao, Ming Li, 00. Gio (005) repored high leel of VIX prediced of fuure marke reurn reersal weakly. Doran e al. (00) showed ha a specific ype of skewness in implied olailiy prediced fuure reurns. Oher relaed research include: Copeland and Copeland (999) repors srong negaie correlaion beween conemporaneous marke reurn and implied olailiy. Banerjee e al. (007) repored ha VIX is negaiely correlaed wih sock marke reurns, a risk facor affecing he sock marke. Bu hese resuls did no esablish clear predicing direcion of implied olailiy as hey are abou he conemporaneous relaionship. 5 curren marke reurn. We aribue his nonlinear relaion o he fac ha marke will change is course when i reaches a reference poin, oherwise, i will coninue is rend. We show analyically ha such reference poin depends on he leel of IV. Therefore, idenical IV alue may forecas fuure reurn differenly depending on how he marke is currenly performing. Our mehod of empirical analysis is inspired by he regime-swiching 3 mehod ha is used exensiely in modeling nonlineariy. Using reurn and opions daa of he S&P 00 from , we examined he relaionship beween fuure weekly marke reurn and he IV on S&P 00 for boh near- or a- he money call and pu opions. We ran simple OLS regressions of fuure marke reurn ono he curren IV condiional on immediae reurn of he S&P 00 index. Our regression resuls confirm he heoreical hypohesis of nonlineariy. To be specific, when curren weekly reurn on S&P 00 is below -%, he regression coefficien is posiie, implying ha a high IV predics a possible fuure marke reersal when curren weekly reurn on S&P 00 is beween -% and -%, he regression coefficien is negaie, implying ha a high IV predics a coninuing fuure marke loss; when curren weekly reurn on S&P 00 declines by less han % or rises, here is no clear relaionship beween leel of he IV and he subsequen marke reurn. In conras o he preious papers, hese resuls sugges here is no a simple, uniform relaionship beween he fuure marke reurn and he condiional olailiy across all marke condiions. Our resul reaffirms ha marke iming decision based on opions implied olailiy is profiable, bu i differs from preious sudies in wo imporan aspecs. Firs, in conras o Gio (005) and Doran e al. (00), our resul srongly suppors he noion ha he condiional olailiy deried from opions predics fuure reurns. Timing based on he IV is poenially profiable. Second, leels of he IV poin o differen direcions of moemen in he fuure reurn under differen sae of he marke. I implies ha under specific condiions, Models in behaioral finance, such as Daniel e al. (998) and Hong and Sein (999), found ha sock marke reurn may behae differenly depending on he sae of he marke. 3 See, for example, James Hamilon (008).

2 Insurance Markes and Companies: Analyses and Acuarial Compuaions, Volume, Issue, 00 reducing marke exposure when he marke olailiy increases could be derimenal o inesors. A smar inesor should in fac add exposure o marke when i jus had a big loss een he olailiy is high. Based on our finding, we explore wo rading sraegies condiional on curren marke reurns and leels of he IV of S&P 00 opions. Gien a high leel of he IV, a Conrarian Trading Sraegy (CTS) bes he marke is abou o reer iself afer i was crashed in he preious week and hus iniiaes a long posiion in he marke. On he oher hand, gien he same high curren leel of he IV, a Momenum Trading Sraegy (MTS) indicaes ha he marke will coninue o decline afer is moderae drop recenly, hus a shor posiion in he marke is esablished. Our performance es shows ha boh sraegies generae ery impressie (risk-adjused) profis. Our finding indicaes ha raders, who desire o explore emperae marke inefficiency, could benefi from rading signal proided by high leels of he IV. We proide a heoreical jusificaion for hese rading sraegies as well.. Daa and mehodology.. Daa. We consruced our sample by using he S&P 00 index in he CRSP daabase and he corresponding opion daa from Opion Merics. We calculaed he weighed aerage of implied olailiy on each Wednesday from June 5, 996 o Sep 0, 008 (634 daa poins oerall). We inesigaed he relaionship beween his measure of IV and he weekly holding period reurn of he S&P 00. We dropped index opions ha rade for $0.05 or less. All opions in he analysis had o hae 00 or more conracs in he open ineress. To aoid noise from far-erm opions and shor-erm opions, we used only opions ha expired in 0-60 days... Implied olailiy. For all opions, we used he implied olailiy of a-he-money and near-he-money calls and pus as well as he olume of he open ineres o consruc he weighed aerage IV. The weighing scheme is suggesed by Laane and Rendleman (976), and Sewar (995). IV N wiv i i i N wi i, We find similar resuls from S&P 500 index and is opions. We do no supply hese resuls here, bu hey are aailable upon reques. In his sense, we beliee our resuls reflec he genuine behaiors of marke reurn and is relaed opions. We use IV of he index opions insead of VIX index as our predicing ariable. VIX is a symmerical olailiy measure in ha i reas sharp moemen in boh direcions equally. Our ineres is mainly o explore he asymmerical response o pas IV in conjuncion wih preious marke moemen. where he weighs w i are he olume of open ineress and IV i is he implied olailiy of a gien opion. We define he moneyness of an opion by he srikeo-spo raio, m i = E i / s. An ATM opion has a srike-o-spo raio beween and including 98% and 0%, while a near-he-money opion has a raio following in eiher 90%, 98% or 0%, 0%. We deleed he obseraions if he srike-o-spo raio is ouside hese ranges..3. Regression model. In his secion, we show analyically ha a nonlinear relaion exiss beween he fuure marke reurn and condiional ariance. The nonlineariy comes from he sae-dependen naure of he regression coefficien. For his reason, we jusify ha he empirical analysis of he relaion mus also be sae-dependen. We sar wih he following basic predicie regression: r, () where r + is he fuure marke reurn. The predicing ariable Var r Iis he condiional ariance of marke reurn r, where I denoes aailable informaion up o ime. In our empirical analysis he measure of he condiional ariance is he IV. The coefficien in equaion () measures he marginal effec of he condiional ariance on he fuure marke reurn. A posiie (negaie) indicaes a posiie (negaie) relaion. The esimaion of is no performed on he whole sample. Insead, we separae our sample ino seeral sub-samples according o curren marke reurns r. To jusify his procedure, we cie a sandard informaion updaing process as in Fama and French (988), Peroba e al. (987), Timmermann, A. (996) and Campbell and Yogo (006). Following he aboe auhors, we posulae ha he marke reurn follows a dynamic facor model: r = x +, x = ax - +w -, N(0, ), w - N(0, ), w () where x is he unobserable laen facor, ha dries he reurn process. The parameer a measures he persisency of he reurn process. In an equilibrium model, he condiional ariance will affec he fuure marke reurn r + hrough marke paricipans updaing mechanism. A ypical updaing process will include he pas esimaes of marke reurn and he condiional olailiy hrough a non-linear funcional form. To see his, we compue he expeced fuure marke reurn condiional on informaion aailable a ime using a recursie Bayesian updaing for- 53

3 Insurance Markes and Companies: Analyses and Acuarial Compuaions, Volume, Issue, 00 mula. We define he expeced reurn condiional on ime informaion as xˆ E x. Then we hae he following recursie formula: E r axˆ, xˆ k axˆ k r where k k /, (3) is called he Kalman filer gain and he condiional ariance iself follows a recursie formula a w. a w Equaions (3) are he Kalman filering ha are deried from he Bayesian updaing. The formula indicaes ha he expeced marke reurn is a weighed aerage of he obsered reurn and preious esimaion of he facor. Thus, he coefficien in equaion () is he deriaie of marke reurn wih respec o he condiional ariance : E r axˆ g k r, (4) a where g a ak is mosly a posiie number. The alue of he Kalman filer gain k lies beween 0 and. Based on equaion (4), i is now sraighforward o obsere ha he sign of depends on he sign of he obsered marke reurn r. ends o be posiie when xˆ (or r ) is a large posiie (or negaie) number, and ends o be negaie or zero, oherwise. For insance, gien a negaie alue of xˆ, he sign of will become posiie if he curren reurn r is a large loss, oherwise i will be negaie. Similarly, he sign of can be analyzed analogously when gien a posiie alue axˆ of xˆ. We recognize ha plays a reference poin for he sign of. Hence, simply running a regression of r + ono he condiional ariance would generae a spurious relaion beween he fuure marke reurn and he condiional ariance. I follows ha we should run simple regressions of equaion () condiional on he alue of curren For deailed deriaion, please referred o Hamilon (994) or Green (008). We erm his phenomenon he reference effec. marke reurn r. Similar o he regime-swiching models, we herefore sored weekly marke reurns ino he following brackes 3 (-, -%), [- %, -%), [-%, 0), [0, %), [%, %) and [%, + ). We erm each bracke as one sae of he reurn, which we denoe as s i. We perform analysis condiional on each sae of reurn. Specifically, equaion () is slighly modified ino he following condiional regression: r, IV if r s i for i =,,,6, where s = (-, -%), s = [-%,- %), ec. Here, we use he implied olailiy IV as he measuremen of he condiional ariance. In addiion, he regression is performed across all moneyness (OTM, ATM and ITM), ypes of opions (i.e. call and pu), and six saes of reurn. A oal of 36 OLS regressions is performed.. Empirical resuls Our empirical resuls include descripie saisics, regression resuls, and performance of rading sraegies based on sae of he index reurn and he IV... Descripie resuls. Table shows he summary saisics for he sample sored by he sae of weekly reurns in he curren week. The firs column liss six groups (saes) of curren reurns. Wihin each group we repor he aerage of reurns on Panel A and implied olailes on Panel B. Aerage IV is repored according o he ype of opions (call pu) in differen moneyness as well as he number of obseraions. Panel A shows ha he aerage marke reurn is abou 0.89% (-0.06%) following a large negaie (posiie) reurn of less (more) han -% (%) in he preious week, which indicaes a reurn reersal. In conras, he marke reurn generally shows momenum when he reurn is beween [-%, -%) or [%, %), because he reurn in he nex week carries he same sign as ha in he pas week. We did no find clear direcion in fuure reurn following a reurn beween -% and % in he curren week. The olailiy saisics in Panel B show ha a pu opion has higher IV han a call opion wih he same srike price all he ime. For example, when he reurn is below -%, he IV of an ou-ofmoney call opion is.55% in he same week, while he IV of an in-he-money pu opion is 3.6%. This finding is consisen wih preious works on implied olailiy skew (e.g., Doran and Kreger 00). 3 These brackes are seleced hrough rials and errors. 54

4 Insurance Markes and Companies: Analyses and Acuarial Compuaions, Volume, Issue, 00 Table. Summary saisics, June 996-Sepember 008 Panel A Panel B. Aerage IV in he curren week If reurn in curren week is Reurn in he nex week OTM call ITM pu ATM call ATM pu ITM call OTM pu <-% Aerage 0.89%.55% 3.6% 4.40% 5.% 7.33% 8.69% n [-%,-%) Aerage -0.3% 6.70% 8.88% 8.69% 9.67%.0%.99% n [-%,0) Aerage 0.% 5.% 8.08% 7.4% 7.98% 0.4%.39% n [0%,%) Aerage -0.07% 4.5% 7.89% 6.3% 7.4% 9.33% 0.64% n [%,%) Aerage 0.5% 5.5% 8.6% 7.53% 8.4% 0.96%.6% n >% Aerage -0.06% 9.48%.4%.9%.57% 5.5% 6.9% n Toal Aerage 0.3% 6.86% 0.06% 9.% 9.83%.5% 3.3% n Noe: Panel A shows he aerage reurn in he following week for each sample. Panel B shows he aerage IV for each group. The size of each sample is indicaed by n. There is a common paern of IV along he dimension of curren reurn for all opions. The IV is he highes when he curren reurn is below -%. I declines as he curren reurn increases o beween 0% and %, and hen i sars o go up. For example, he olailiy has is highes alue of 8.69% for he ou-of-money pu opion when he reurn is below -%. The same IV drops o 0.63% when he curren reurn is beween 0 o %, and i creeps back o 6.8% when he marke had a reurn of more han %. This U-shape moemen of he IV along he dimension of reurn indicaes ha he IV is high when he marke has more exreme reurns, where inesors may reflec on more uncerainy... The relaion beween IV and marke reurn. We are ineresed in finding how he IV predics he fuure index reurn. We condiioned our OLS regression on each of he six differen saes of he marke reurn. We repor he resuls for boh call and pu opions in separae ables. Since here are hree ypes of moneyness for each opion, we hae 8 regressions in each able. Table repors esimaion resuls for he 8 condiional regressions for call opions. Each nonparenhesized number represens he esimae of slope for one ype of moneyness condiional on one sae of reurn. We suppress esimaes of inerceps. Column A indicaes ha he subsequen reurn would sar o increase if he reurn has dropped more han -%. The size of increase is proporional o he IV because he regressions generae a posiie coefficien wih more han 99% of confidence leel. The IV of ou-of-money calls has he larges coefficien wih a alue of Since he aerage IV is abou 0%, i would add abou 3.76% o he nex reurn. Surprisingly, column B shows ha he subsequen reurn will coninue o decrease if he curren week s reurn has dropped by more han % bu no greaer han %. The IV of ou-of-money calls predics mos decline wih a negaie coefficien of Table. Regression coefficiens of call opion IV on S&P 00 index reurn. IV A. r(-,-%) B. r[-%,-%) C. r[-%,0) D. r[0,%) E. r [%,%) F. r[%,) OTM call 0.88*** -0.3*** (0.0508) (0.0478) (0.0385) (0.036) (0.046) (0.0476) ATM call 0.5*** -0.0** (0.0457) (0.0449) (0.0340) (0.037) (0.0369) (0.044) ITM call 0.44** -0.00* (0.0555) (0.0546) (0.0346) (0.0305) (0.038) (0.040) Noe: The model is r c IV, where IV is he implied olailiy from one of he following call opions: ou-of-he-money, a-hemoney and in-he-money calls. Each number wihou parenheses is an esimae of for one of he 8 OLS regressions for call opion. The numbers in parenheses under esimaes are he sandard deiaions of he indicaed ariable. The sample of each regression is seleced according o he reurn in preious week. The significance leel of esimaes is indicaed by he number of aserisks: % (***), 5% (**) and 0% (*). All sample size of each regression aries from 6 o 4. Inerceps are no repored here. The suppressed inerceps are saisically negaie relaed o column A and saisically posiie relaed o column B wih a leas 0% significance leel. All oher inerceps are saisically insignifican. 55

5 Insurance Markes and Companies: Analyses and Acuarial Compuaions, Volume, Issue, 00 Anoher conrasing resul is ha when he marke reurn is greaer han -%, he IV has no predicie power across all ypes of moneyness because all esimaes in column C-F are insignifican a een 90% confidence leel. 56 The resuls indicae ha informaion conained in opions is useful for predicing he fuure reurn only when he weekly marke reurn has dropped by more han %, bu he prediced direcion is compleely opposie depending on wheher he loss is more han % or no. Table 3. Regression coefficiens of pu opion IV on S&P 00 index reurn. IV A. r(-,-%) B. r[-%,-%) C. r[-%,0) D. r[0,%) E. r[%,%) F. r[%,) OTM pu 0.6*** -0.0*** (0.0475) (0.043) (0.033) (0.035) (0.0367) (0.0455) ATM pu 0.6*** -0.7*** (0.0496) (0.0437) (0.034) (0.0330) (0.0370) (0.0469) ITM pu 0.6*** -0.58*** * (0.057) (0.050) (0.0447) (0.0645) (0.065) (0.0709) Noe: See he noes o earlier ables for ariable definiion and model informaion. Sample size of each regression aries from 6 o 4. Table 3 repors esimaes of he 8 OLS regressions for pu opions. They are similar o hose for he call opions. Column A repors ha if he S&P 00 index drops by more han % in curren week, he IV predics an increase in he index nex week by a rae of 0.6 for each moneyness. Column B shows ha when he index loses are beween % and %, he IV predics ha he marke will coninue he losing sreak for he nex week because he regression coefficiens are negaie for all IV across differen moneyness. All hese esimaes are significanly differen from zero a a 99% confidence leel. Similar o he resuls for call opions, he IV does no hae significan predicing power on he fuure reurn when he marke reurn is greaer han -% in he curren week. The esimaes are repored in columns C o F..3. Performance of conrarian and momenum rading sraegies. According o our findings, he naural porfolio sraegy will be iming buy or sell based on he marke reurn and he leel of he IV. One implicaion of he finding is ha here is a significan posiie reurn indicaed by he IV afer a more han % drop in he marke reurn. A conrarian sraegy could buy a his marke downurn and profi on he subsequen reersal. On he oher hand, he momenum sraegy should shor he marke index when he marke reurn drops more han %, bu no greaer han %. Bu boh sraegies should be execued only when he IV is high. Here, we proide an analyical argumen why he sraegies would profi under high IV..4. Inuiion for he sraegies. Imagine ha an inesor wans o maximize her sandard uiliy funcion E U R, n i where R wr iri is he porfolio reurn, which includes he equiy marke reurn r. Under sandard assumpions and he meanariance analysis, he opimal weigh o he equiy marke is equal o w E r, which depends on he condiional ariance. Since obiously he weigh o he equiy marke has a posiie correlaion wih he marke index, he condiional olailiy will affec he marke index oo. We firs examine how he condiional ariance affecs he inesor s allocaion in he equiy marke. Uilizing he definiion of in equaion (4), he parial deriaie of opimal weigh w wih regard o is : w E r. (5) Equaion (5) shows ha he adjusmen of he posiion Er w in equiy depends on and. Our preious resul indicaes ha can be eiher posiie or negaie, so he inesor s change of he weigh on equiy aries depending on he sign of. When he curren marke was in a loss sae of [-%, -%], we know ha is negaie (from our empirical resul). Applying his knowledge in equaion (5), we find ha inesors will respond o reduce heir posiion in equiy because w < 0 now. This implies ha furher price decline is more likely in ha equiy marke and he bes acion is o follow he Momenum Trading Sraegy, i.e., selling a he marke mild drop o aoid furher price decline. On he oher hand, when he marke incurred a deep loss (a drop of % or more in a week), he posiie sign of from he regression may lead o an increasing size of he weigh on equiy in equaion (5) if is large Busse (999) has a similar formula in sudying he olailiy iming of muual funds.

6 E r enough o offse, which creaes a profi opporuniy for following he Conrarian Trading Sra- egy, i.e., buying a he marke dip o exploi possible price reersal. To furher examining equaion (5), we found ha Er he alue of becomes relaiely small a high condiional olailiy, which minimizes is impac on he inesor s adjusmen of he equiy weigh. Therefore, he change in he equiy posiion w relies more on he sign of a high marke olailiy (IV), which proides a clearer signal for inesors o follow eiher MTS or CTS..5. Performance saisics. We hae jus shown ha he MTS or CTS should be carried ou under he condiion of a high IV. To find a sensible gauge for a high leel of he IV, we use he hisorical disribuion of he IV for each opion. We consider he IV as high if i is aboe he cuoff of is 75% percenile of is hisorical disribuion. Table 4 displays he cuoffs of 75% percenile of he IV for each opion. To aoid being arbirary on he cuoff poins, we perform similar analysis Insurance Markes and Companies: Analyses and Acuarial Compuaions, Volume, Issue, 00 for cuoffs of oher highs for robus check in he nex subsecion. Table 4. The 75% percenile of implied olailiy OTM call ITM pu ATM call ATM pu ITM call OTM pu Table 5 repors weekly reurns from conrarian rading condiional on reurn being less han -%. The sraegy generaes a weekly reurn of a leas.4% if he IV falls in is op quarer of is hisorical disribuion. Similar resul holds under oher measures of he IV. The sandard deiaions of hese weekly reurns under differen IV are around 3%. Gien a risk-free rae of 6% annually, he Sharpe raio is beween.87 and 3.56, which is ery impressie compared o any oher asse in he marke. Gien he same risk-free rae of 6%, a ypical Sharpe raio for S&P 500 is abou 0. if i aerages 0% reurn wih a 5% sandard deiaion annually. Alhough he coefficien is posiie, we find ha he conrarian sraegy does no generae significan profi when he IV is no high (below is 75% percenile cuoff). I may indicae he posiie effec of IV on reurn is no enough o oercome he reference poin, which would coninue is downside pull on he reurn. Table 5. Time + reurns from conrarian rading when he ime marke reurn was in (-, -%) IV high IV no high Type of opion Mean Sandard de Sharpe raio Sample size Mean Sandard de Sharpe raio Sample size OTM call.7% 3.% %.00%.3 34 ATM call.5%.98% %.03% ITM call.4% 3.7% %.00% OTM pu.6% 3.6% %.08% ATM pu.53% 3.33% %.00% ITM pu.57% 3.67% %.% Noe: The Sharpe raio is calculaed assuming 6% risk-free rae. IV is high if i is aboe is corresponding cuoff in able 4, oherwise i is no high. Table 6 repors reurns from momenum rading ha is condiional on ha he marke reurn is beween -% and -%. Like in Table 5, he highes reurn is achieed when he IV is in he highes quarile (aboe is 75% percenile). The reurn ranges from.% o.45%. The sandard deiaion ranges from 3.6% o 3.74%. The Sharpe raio would range from o.5. Again reurns from oher momenum sraegies are much less when he IV is no high. Table 6. Time + reurns from momenum rading when he ime marke reurn was in [-%, -%] IV high IV no high Type of opion Mean Sandard de Sharpe raio Sample size Mean Sandard de Sharpe raio Sample size OTM call.0% 3.66% %.78% ATM call.06% 3.6% %.78% ITM call.00% 3.6% %.84% OTM pu.39% 3.49% %.70% ATM pu.6% 3.65% %.7% ITM pu.45% 3.74% %.83% See he noes o earlier able for definiion of IV - high. The empirical disribuion of hisorical IV is quie sable oer ime. 57

7 Insurance Markes and Companies: Analyses and Acuarial Compuaions, Volume, Issue, 00 We also experimened on rading under oher circumsances, where he reurn falls ino oher brackes. The reurns are all discouraging. We repored fuure reurns when he curren reurn is aboe -% in Table 7. Reurns from eiher sraegy show insignifican or ambiguous resuls. Table 7. Time + reurns from buy when he ime marke reurn was greaer -% IV high IV no high Type of opion Mean Sandard de Sharpe raio Sample size Mean Sandard de Sharpe raio Sample size OTM call -0.06% 3.35% %.78% ATM call 0.% 3.6% %.8% ITM call 0.03% 3.09% %.8% OTM pu -0.07% 3.8% %.8% ATM pu -0.04% 3.8% %.8% ITM pu 0.0% 3.79% %.04% Noe: All reurns come from buy posiion. See he noes o earlier able for definiion of IV - high..6. Performance under differen high IVs. We proide resuls analogous o Table 5 and Table 6 under differen leels of high IV. To sae space, we only repor he resuls when he IV is high. We also suppressed sandard deiaions. We define IV as high if i is aboe he corresponding cuoff poins. They are shown in Table 8 and Table 9. Table 8 repors conrarian reurns, when he IV is high. The resuls resemble o hose in Table 5 in ha he weekly reurn is aboe % wih ery impressie Sharpe raios. We also noe ha he aerage reurn and Sharpe raio generally increase as we push up he cuoff for high IV. Bu he number of weeks for conrarian rading also decreases a he same ime. Therefore, i is no necessary a good idea o increase he cuoff. A good balance beween a higher aerage reurn and a good size of sample would need more inesigaion. A similar siuaion is in Table 9 compared o Table 6, which repors reurns from he momenum rading. Table 8. Time + reurns from conrarian rading when he marke reurn was in (-, -%) a ime Type of Mean Share raio Sample size Mean Share raio Sample size Mean Share raio Sample size opion IV High (>50%) High (>67%) High (>80%) OTM call.04% % % ATM call.06%. 76.5% % ITM call.07% % % OTM pu.0% % % ATM pu.00% % % ITM pu.% % % High (>87.5%) High (>90%) High (>95%) OTM call.6% % % ATM call.89% % % 4. 8 ITM call.4% %.69.69% OTM pu.06% % % ATM pu.90% % % ITM pu.39% % % Noe: The number in each pair of parenheses is he percenage cuoff for a high olailiy from he hisorical disribuion of he IV. Table 9. Time + reurns from momenum rading when he marke reurns was in (-%, -%) a ime Type of Mean Share raio Sample size Mean Share raio Sample size Mean Share raio Sample size opion IV High (>50%) High (>67%) High (>80%) OTM call 0.40% %.7 5.3%.3 3 ATM call 0.9% % %.8 ITM call 0.55% %.64.8% 3.44 OTM pu 0.7% % %.65 3 ATM pu 0.74% %..3%. ITM pu 0.98% % %.58 9 High (>87.5%) High (>90%) High (>95%) OTM call 3.4% % % 5. 3 ATM call 3.38% % %

8 Insurance Markes and Companies: Analyses and Acuarial Compuaions, Volume, Issue, 00 Table 9 (con.). Time + reurns from momenum rading when he marke reurns was in (-%, -%) a ime Type of Mean Share raio Sample size Mean Share raio Sample size Mean Share raio Sample size opion IV High (>87.5%) High (>90%) High (>95%) ITM call.79% % % 5. 3 OTM pu.46% % % 35.4 ATM pu.96% % % 5. 3 ITM pu 3.9% % % 35.4 See noes in preious ables. Conclusion In his paper, we find an asymmeric paern for he IV of he S&P 00 opions as a predicor of he fuure marke reurn. Praciioners hae long suspeced ha a high IV signals an oersold marke. Our finding suppors he alidiy of such claim only when he weekly marke reurn drops by more han % and he IV is a a high leel. We beliee his reersal phenomenon is robus because i has occurred a leas 43 weeks during In conras o preious sudies, we also discoered ha when he loss in he marke is moderae (i.e., weekly loss beween % and %), he IV in fac predics a coninual loss. References Researchers hae focused on he link beween he implied olailiy and he fuure realized olailiy. Few sudies deal wih he possible relaionship beween he implied olailiy and fuure reurns. We hope our finding can make up some of he missing par in he empirical research on his aspec. In addiion, he sandard finance heory canno explain readily why a high IV should predic significan marke reurns. The predicabiliy of reurn and abnormal reurns from our es sraegies is agains he hypohesis of marke efficiency. We hope ha fuure sudy may reconcile his anomaly wih a judicious heory of finance.. Barcus, D.K and A. W. Gregory, 993. Theoreical relaions beween Risk Premiums and Condiional Variances, Journal of Business and Economic Saisics,, Banerjee, Prihiraj S., James S. Doran, Daid R. Peerson. Implied olailiy and fuure porfolio reurns, Journal of Banking & Finance, 007, 3, Busse, Jeffrey A. Volailiy Timing in Muual Funds: Eidence from Daily Reurns, Reiew of Financial Sudies, Winer 999, Vol., No. 5, Campbell, John Y. and Moohiro Yogo, 006. Efficien ess of sock reurn predicabiliy, Journal of Financial Economics 8, Copeland, Maggie M. and Thomas E. Copeland. Marke Timing: Syle and Size Roaion Using he VIX, Financial Analys Journal, March/April 999, Daniel, K., Hirshleifer, D., Subrahmanyam, A., 998. Inesor psychology and inesor securiy marke underand oerreacion, Journal of Finance, 53, Doran, James S, Kein Krieger. Implicaions for Asse Reurns in he Implied Volailiy Skew, Financial Analyss Journal, Jan/Feb 00. Vol. 66, Iss., p. 65 ( pages). 8. Eraker, Bjørn, Michael Johannes and Nicholas Polson. The Impac of Jumps in Volailiy and Reurns, The Journal of Finance, Vol. 58, No. 3 (Jun., 003), pp Fama, Eugene and Kenneh French. Permanen and Temporary Componens of Sock Prices, JPE, Vol. 6, Issue, Apr 988, Fleming, Jeff, Chris Kirby and Barbara Osdiek, 00. The Economic Value of Volailiy Timing Using Realized Volailiy, Working paper, Rice Uniersiy.. Gio, Pierre, Implied olailiy indices as leading indicaors of sock index reurns?, Journal of Porfolio Managemen, Vol. 3, pp. 9-00, Graham, John and Campbell Harery, 997. Grading he Performance of Marke-Timing Newsleers, Financial Analyss Journal, Green, W. Economeric Analysis, 5 h ED, Prenice Hall, Hamilon, J. Time Series Analysis, Princeon Uniersiy Press, Hamilon, J. Regime-Swiching Models, The New Palgrae Dicionary of Economics, Hong, H., Sein, J., 999. A unied heory of underreacion, momenum rading, and oerreacion in asse markes, Journal of Finance, 54, Johannes, Michael and Nicholas Polson, Volailiy Timing and Porfolio Reurns, Working Paper, Uniersiy of Chicago. 8. Koop, Gary. Bayesian Economerics, Wiley, Laan, G, Henry A., and Richard J. Rendleman, Jr Sandard Deiaions of Sock Price Raios Implied in Opion Prices, The Journal of Finance, Vol. 3, No. (May):

9 Insurance Markes and Companies: Analyses and Acuarial Compuaions, Volume, Issue, Leland, Hayne, 999. Beyond Mean-Variance: Performance in a Nonsymmerical World, Financial Analyss Journal, Li, Ming. Informaion Precision and Price Momenum, Working Paper, San Francisco Sae Uniersiy, Mayhew, Sewar. Implied Volailiy, Financial Analys Journal, July-Augus, 995, Peroba, James and Summers, Lawrence. Mean reersion in sock reurns: eidence and Implicaions, Journal of Financial Economics, Vol., Issue, Oc 988, Scruggs, J.T., 998. Resoling he Puzzling Iner-emporal Relaion beween he Marke Risk Premium and Condiional Marke Variance: A Two-Facor Approach, Journal of Finance, 53, Timmermann, A., 996. Excess Volailiy and Predicabiliy of Sock Prices in Auoregressie Diidend Models wih Learning, Reiew of Economic Sudies, 63, Whielaw, R.F. 000, Sock Marke Risk and Reurn: An equilibrium Approach, Reiew of Financial Sudies, Vol. 3, No. 3, pp Appendix. Deriaion of he regression coefficien The coefficien is defined as E r ˆ E r xˆ ˆ ˆ a x x a r ˆ ax a ak r a. From equaion () we find ha E r E ax w ae x ax. Therefore, combining wih he formula of Kalman filering, he firs deriaie is. w a a w Gien ha, we hae he following deriaie, w, where a w is a w ypically inerpreed as he singal-o-noise raio in he reurn daa. The ypical alue of is in he range of [00,00] (see Koop, 003; Li, 008). I can be shown ha 0, since is a weighed aerage of w and. We find ha is a ery large posiie number. A simulaion for a=0.9 shows ha he range of is beween 500 and 000. xˆ ˆ x Assume he firs-order effec of on x ˆ conerges o consan as ime approaches he infiniy, i.e.. Collecing erms in he aboe equaion, we arrie a simplified formula for he deriaie: Er axˆ a g k r, where g a ak since is ery large, g is mosly posiie gien ha a and k are boh beween 0 and. 60