ROBERT E. WHALEY * Undersanding VIX ABSTRACT In he recen weeks of marke urmoil, financial news services have begun rouinely reporing he level of he CBOE s Marke Volailiy Index or VIX, for shor. While his new pracice is healhy in he sense ha invesors are asking for more informaion in helping o assess he sae of he curren economic environmen and o guide hrough urbulen waers, i is imporan o undersand exacly wha he index means in order o fully capure is usefulness o he marke and o avoid misundersanding and misconcepion. The purpose of paper is o describe he VIX and is hisory and purpose, and o explain how i fis wihin he array of indexes ha help describe where he economy sands relaive o oher poins in recen decades. Firs draf: November 6, 2008 Revised: December 5, 2008 *Valere Blair Poer Professor of Finance, The Owen Graduae School of Managemen, Vanderbil Universiy, 401 21 s Avenue Souh, Nashville, TN 37203. Telephone: (615) 343-7747, Fax: (615) 376-8879, Email: whaley@vanderbil.edu. Elecronic copy available a: hp://ssrn.com/absrac=1296743
Undersanding VIX In he recen weeks of marke urmoil, financial news services have begun reporing he VIX wih increasing regulariy. This new baromeer of invesor fear he CBOE s Marke Volailiy Index is claimed o have reached unprecedened levels and is now, in fac, causing sock marke volailiy. 1 While such misconcepions will, undoubedly, coninue o flow, his aricle hopes o sem he ide by explaining wha VIX is and is no, why i was creaed, wha causes i o move, and why i should be an imporan piece o invesors. Among he lessons o be learned are ha he VIX is no new, is no a unprecedened levels, and does no cause marke volailiy. I. The VIX To begin, VIX is an index, like he Dow Jones Indusrial Average (DJIA), compued on a real-ime basis hroughou each rading day. The only meaningful difference is ha i measures volailiy and no price. VIX was inroduced in 1993 wih wo purposes in mind. Firs, i was inended o provide a benchmark of expeced shor-erm marke volailiy. To faciliae comparisons of he hen-curren VIX level wih hisorical levels, minue-by-minue values were compued using index opion prices daing back o he beginning of January 1986. This was paricularly imporan since documening he level of marke anxiey during he wors sock marke crash since he Grea Depression he Ocober 1987 Crash would provide useful benchmark informaion in assessing he degree of marke urbulence experienced subsequenly. Second, VIX was inended o provide an index upon which fuures and opions conracs on volailiy could be wrien. The social benefis of rading volailiy have long been recognized. The Chicago Board Opions Exchange (CBOE) launched rading of VIX fuures conracs in May 2004 and VIX opion conracs in February 2006. In aemping o undersand VIX, i is imporan o emphasize ha i is forward-looking, measuring volailiy ha he invesors expec o see. I is no backward-looking, measuring volailiy ha has been recenly realized, as some commenaors someimes sugges. Concepually, VIX is like a bond s yield o mauriy. Yield o mauriy is he discoun rae ha 1 A couple of recen quoes include Now we re in unchared erriory in Volailiy: Measure Could Signal a Boom (Philadelphia Inquirer, November 1, 2008) and The VIX is a self-fulfilling prophecy in On Wall Sree, Eyes Turn o he Fear Index (New York Time, Ocober 20, 2008). 1 Elecronic copy available a: hp://ssrn.com/absrac=1296743
equaes a bond s price o he presen value of is promised paymens. As such, a bond s yield is implied by is curren price and represens he expeced fuure reurn of he bond over is remaining life. In he same manner, VIX is implied by he curren prices of S&P 500 index opions and represens expeced fuure marke volailiy over he nex 30 calendar days. II. VIX calculaion The VIX was inroduced by Whaley (1993). The original index was based on he prices of S&P 100 (icker symbol OEX ), no S&P 500 (icker symbol SPX ), index opion prices. The reason was simple. A he ime, OEX opions were he mos acively-raded index opions in he U.S., accouning for 75% of he oal index opion volume in 1992. 2 In conras, he SPX opion marke was abou one-fifh as acive, accouning for only 16.1% of volume. Criical o he imeliness and usefulness of any implied volailiy index (of which VIX is only one) is o have i based on prices from a deep and acive index opion marke. In 1992, he dominan source was he OEX opion marke. A second feaure of he original VIX was ha i was based on he prices of only eigh a-he-money index calls and pus. 3 Again, his was reasonable. Of he available opion series a he ime, a-he-money opions were by far he mos acively raded. Opions wih exercise prices were away from he curren sock index level were less acively raded and frequenly had sale price quoes and relaively wide bid/ask spreads. Including such quoes in he real-ime compuaion of he VIX would reduce is imeliness and accuracy. Over he years since is incepion, he srucure of index opion rading in he U.S. changed in wo fundamenal ways. Firs, he SPX opion marke became he mos acive index opion marke in he U.S. Currenly, SPX opions rade abou 12.7 imes as frequenly as OEX opions. 4 Exacly why he rading volume shifed from one marke o he oher is unclear. Conribuing facors include he fac ha he S&P 500 index is beer known, fuures conracs on he S&P 500 are acively raded, and S&P 500 opion conracs are European-syle (i.e., exercisable only a expiraion), making hem easier o value. 5 On he oher side of he coin, he average daily rading volume in OEX opions in 2008 was less han half of wha i was 16 years earlier. Irrespecive of he reason(s), a imely and meaningful implied volailiy index requires 2 See Whaley (1993, p.72). 3 The deails of he original index compuaion are provided in Whaley (1993, pp. 80-82). 4 This figure is based on he firs en monhs of rading in 2008. 5 OEX opions, on he oher hand, are American-syle and can be exercised a any ime during he opion s life. 2
prices from an acive underlying index opion marke. The OEX opion marke had been supplaned by he SPX opion marke, and i was ime for a change. Second, rading moives of marke paricipans in index opion markes changed. In he early 1990s, boh index calls and index pus had equally imporan roles in invesor rading sraegies. Trading volumes were balanced in 1992, wih OEX calls having an average daily rading volume of 120,475 and OEX pus an average daily rading volume of 125,302. Over he ensuing years, he index opion marke became dominaed by porfolio insurers, who rouinely buy ou-of-he-money and a-hemoney index pus for insurance purposes. During he firs en monhs of 2008, for example, he average daily volume of SPX pus was 909,748 conracs, over 72% more han he 525,460 for SPX calls. Indeed, as Bollen and Whaley (2004) show, he demand o buy ou-of-he-money and a-he-money SPX pus is a key driver in he movemen in SPX implied volailiy measures such as VIX. On Sepember 22, 2003, he CBOE changed he VIX calculaion o accoun for boh of hese fundamenal changes in index opion marke srucure. Firs, hey began o use SPX raher han OEX opion prices. Second, hey began o also include ou-of-he-money opions in he index compuaion since ou-of-he-money pu prices, in paricular, conain imporan informaion regarding he demands for porfolio insurance and, hence, marke volailiy. Including addiional opion series also helps make he VIX less sensiive o any single opion price and hence less suscepible o manipulaion. 6,7 I is worh noing ha he change from OEX opion prices o SPX opion prices had lile o do wih he reurn/risk properies of he indexes hemselves. For all inens and purposes, he S&P 100 and S&P 500 index porfolios are perfec subsiues. 8 Over he period January 1986 hrough Ocober 2008, he mean daily reurns of he S&P 100 and S&P 500 were nearly idenical, 0.0263% and 0.0266%, respecively, and he sandard deviaions of S&P 100 daily reurns was only slighly higher han he S&P 500 reurns, 1.182% and 1.138%, respecively. 6 From he perspecive of rading derivaives conracs on he VIX, he change in mehodology also provided a means of rading VIX by passively using SPX opion conracs. This provides marke makers in VIX fuures and opions wih a less expensive means of hedging heir invenory and promoes narrower bid/ask spreads in he VIX fuures and opions markes. Under he original version of VIX, dynamic hedging was required. See Carr and Madan (1998) and Demeerfi, Derman, Kamal, and Zou (1999). 7 The deails of he revised VIX compuaion are provided in CBOE (2003). An Excel-based spreadshee for compuing he VIX is provided in Whaley (2006, pp. 553-562). 8 Boh he S&P 100 and he S&P 500 are marke capializaion-weighed sock indexes. As of Ocober 31, 2008, all S&P 100 socks are conained wihin he S&P 500 index porfolio and accoun for 62.46% of he S&P 500 s oal marke capializaion. The 34 highes marke cap socks in he S&P 500 are also he 34 highes marke cap socks in he S&P 100. Of he 100 highes marke cap socks in he S&P 500, 70 are from he S&P 100. 3
The correlaion beween heir daily reurns was 0.9898. The near perfec correlaion beween he reurn series implies ha, holding oher facors consan, OEX and SPX opions are equally effecive from a risk managemen sandpoin. Bu, oher facors are no consan. From he sandpoin of mainaining he VIX as a imely and accurae reflecion of expeced sock marke volailiy wha maers is he deph and liquidiy of he index opion marke. The decision o urn o SPX opion prices was warraned. The Invesor Fear Gauge The VIX has been dubbed he invesor fear gauge. While volailiy echnically means unexpeced moves up or down, he S&P 500 index opion marke has become dominaed by hedgers who buy index pus when hey are concerned abou a poenial drop in he sock marke. Buying insurance is nohing new. People rouinely buy fire insurance as a means of insuring heir home value in he even of a fire. If he chance of a fire in your neighborhood rises, chances are ha your insurance agen will charge more for coverage. The same is rue for porfolio insurance. The more invesors demand, he higher he price. VIX is an indicaor ha reflecs he price of porfolio insurance. III. VIX hisory Ofenimes individuals seek o find a precise meaning o an index level. The real benefi from an index, however, comes from comparing is curren level o some hisorical benchmark(s). Consider, for example, he fac ha, on Ocober 24, 2008, he Dow closed a 8378.95. Which maers more he fac ha he level equals he sum of he prices of Dow 30 socks (1052.00) divided by is curren divisor (0.1255527090), or he fac ha he Dow exceeded 11000 only a monh earlier? Mos people hone in on he laer and hink abou is relevance o heir porfolio holdings. Thus, o gauge he normal behavior of he VIX, we should look o is hisory. 9 Figure 1 shows week-ending levels of he S&P 500 and he VIX from he beginning of January 1986 9 The CBOE changed he composiion of VIX on Sepember 22, 2003. For he period, January 2, 1986 hrough Sepember 19, 2003, i was based on S&P 100 index opion prices. Aferward, i was based on S&P 500 index opion prices. Earlier we showed ha he S&P 100 and S&P 500 index porfolios were virually perfec subsiues for one anoher, so using he VIX hisory based on S&P 100 prices unil Sepember 22, 2003 (i.e., he cleaner, more accurae hisorical series) and hen he VIX hisory based on SPX opion prices is a sensible way o develop a full VIX hisory. Anoher, perhaps more accurae, approach is o aemp o address he scale difference in he volailiies of he wo volailiy series. Earlier we showed ha he daily sandard deviaion of he S&P 500 index was 0.011378 4
hrough Ocober 31, 2008. Several observaions are noeworhy. Alhough no obvious from he weekly figure, he level of VIX reached is record high during he Ocober 19, 1987 marke crash. This was he only ime VIX ever exceeded a level of 100. By he week s end, VIX fell o a level slighly below 90, however, i coninued o persis a abnormally high levels in he ensuing weeks. Anoher ineresing phenomenon shown in he figure is ha he VIX frequenly spikes upward. Naurally, he marke crash in Ocober 1987 is an example, however, he jump in Ocober 1989 is he mini-crash resuling from he UAL resrucuring failure, he jump in mid- 1990 occurred when Iraq invaded Kuwai, and he jump in early 1991 corresponds o he aack on Iraq by he Unied Naions forces. Then wo sharp spikes occurred one in Ocober 1997, and one in Ocober 1998. The Ocober 1997 spike occurred following a sock marke sell-off in which he Dow fell 555 poins. The Ocober 1998 spike occurred in a period of general nervousness in he sock marke. In he afermah of each spike, he VIX reurns o more normal levels. Hopefully, he Ocober 2008 spike is no excepion. Finally, alhough he weekly closing levels of he VIX and he S&P 500 index appear o spike in opposie direcions, here are also imes when a run-up in sock prices is accompanied by a run-up in volailiy. In January 1999, for example, he VIX was rising (i.e., invesors were becoming more nervous) while he level of he S&P 500 index was rising. The same paern appears in he firs wo monhs of 1995, June and July of 1997, and December 1999. Clearly, invesors can become nervous even during marke advances. while he daily sandard deviaion of he S&P 100 index was 0.011822. The fac ha he S&P 500 volailiy is only 96.24% of he S&P 100 volailiy reflecs he fac ha he S&P 500 has marginally lower risk. Consequenly, we adjus he pre-sepember 22, 2003 levels of he OEX opion based VIX o reflec his scale difference. To esimae he adjusmen, we regress he daily VIX based on SPX opions prices on he daily VIX based on OEX opion prices during he period Sepember 22, 2003 hrough Ocober 31, 2008, suppressing he inercep erm in he regression. The esimaed slope coefficien is 0.9727. To creae he SPX opion-based VIX prior o Sepember 22, 2003, we simply scale he OEX opion based VIX by a facor of 0.9727. Wih or wihou he adjusmen, he qualiaive inerpreaions of he VIX hisory are unaffeced. 5
Figure 1: Friday closing levels of he S&P 500 index and he VIX during he period January 3, 1986 hrough Ocober 31, 2008. IV. VIX relaion o he sock marke The fac ha he VIX spikes during period s of marke urmoil is why i has become known as he invesor fear gauge. Two forces are a play. If expeced marke volailiy increases (decreases), invesors demand higher (lower) raes of reurn on socks, so sock prices fall (rise). This suggess he relaion beween rae of change in VIX should be proporional o he rae of reurn on he S&P 500 index. Bu, he relaion is more complicaed. Earlier we argued and documened ha increased demand o buy index pus affecs he level of VIX. Hence, we should expec o find ha he change in VIX rises a a higher absolue rae when he sock marke falls han when i rises. To es his proposiion, we regress he daily rae of change of he VIX, of change of he S&P 500 porfolio, RVIX, he rae RSPX, and he rae of change of he S&P 500 porfolio condiional on he marke going down and 0 oherwise, RSPX, ha is, RVIX = β + β RSPX + β RSPX + ε 0 1 2 6
If our proposiion is rue, he inercep erm should no be significanly differen from 0, and he slope coefficiens should be significanly less han 0. As i urns ou, our predicions are rue. The esimaed relaion beween he rae of change of VIX and he rae of change in SPX is RVIX = 0.004 2.990RSPX 1.503RSPX, where he number of observaions used in he esimaion is 5,753 and he regression R-squared is 55.7%. Excep for he inercep, all regression coefficiens are significanly differen from zero a he 1% level. The esimaed inercep in he regression is 0.004, and he inercep is no significanly differen from 0. This means ha if he SPX does no change over he day, he rae of change in VIX should be negligible. This is no surprising. While he value of socks is expeced o grow hrough ime in order o compensae invesors for puing heir capial a risk, volailiy is no. Volailiy ends o follow a mean-revering process: when VIX is high, i ends o be pulled back down o is long-run mean, and, when VIX is oo low, i ends o be pulled back up. The esimaed inercep reflecs he absence of deerminisic growh. The esimaed slope coefficiens are boh negaive and significan, and clearly reflec no only he inverse relaion beween movemens in VIX and movemens in he S&P 500 bu also he asymmery of he movemens brough abou by porfolio insurance. The way o inerpre he coefficiens is as follows. If he SPX rises by 100 basis poins, he VIX will fall by ( ) RVIX = 2.990.01 = 2.99%. On he oher hand, if he S&P 500 index falls by 100 basis poins, VIX will rise by ( ) ( ) RVIX = 2.990.01 1.503.01 = 4.493%. Because of he demand for porfolio insurance, he relaion beween raes of change in he VIX and he SPX is asymmeric. VIX is more a baromeer of invesors fear of he downside han i is a baromeer of invesors exciemen (or greed) in a marke rally. I is imporan o noe, however, his evidence merely documens correlaion and is no inended o express causaliy. 7
V. VIX normal range Aside from looking a VIX levels, we can aemp o characerize wha is normal and abnormal behavior. Table 1 does so in a probabilisic sense. Over is enire hisory, he median daily closing level of VIX is 18.88. 50% of ime VIX closed beween 14.60 and 23.66 (a range of 9.06 poins), 75% of he ime VIX closed beween 12.04 and 29.14 (a range of 17.10 poins), and 95% of he ime VIX closed beween 11.30 and 37.22 (a range of 22.92 poins). Table 1 also shows a grea of variaion in wha is considered normal from year o year. In 1986, for example, he median daily closing level of VIX was abou 19.25. During ha same year, he closing levels were abou 18.06 and 21.07 abou half he ime and beween 16.92 and 24.24 abou 90% of he ime. The wides range experienced is 2008, wih he VIX closing beween 18.16 and 63.31 (a range of 45.15 index poins) abou 90%. The second wides range is in 1987 he year of he sock marke crash. The 5% and 95% perceniles indicae ha he range of daily VIX levels was from 16.64% o 54.11, or 3,474 basis poins. Table 1: Normal ranges for daily levels of VIX over sample period, January 1986 hrough Ocober 2008 and by year. No. of Year obs. 5.0% 10.0% 25.0% 50.0% 75.0% 90.0% 95.0% All 5,754 11.30 12.04 14.60 18.88 23.66 29.14 34.22 1986 252 16.92 17.34 18.06 19.25 21.07 23.64 24.24 1987 253 16.64 17.28 20.85 22.66 26.81 46.25 54.11 1988 253 17.44 18.11 20.35 24.06 27.21 34.20 36.16 1989 252 15.51 15.90 16.47 17.30 18.22 20.60 22.59 1990 253 16.55 17.32 18.31 21.16 26.11 28.96 30.49 1991 251 14.99 15.29 16.02 17.29 19.13 21.87 24.46 1992 254 12.18 12.73 13.36 14.76 15.98 17.33 17.96 1993 251 10.43 10.92 11.40 12.27 13.03 14.05 14.38 1994 252 10.26 10.49 11.29 12.80 14.47 15.63 16.07 1995 252 10.71 11.00 11.51 12.29 13.18 13.80 14.15 1996 254 13.43 14.70 15.72 16.78 18.16 19.39 20.45 1997 253 19.92 20.17 21.11 22.20 24.64 27.80 30.36 1998 252 18.06 18.82 20.43 22.61 27.67 36.37 41.49 1999 252 19.70 20.73 22.39 24.29 26.59 28.73 30.34 2000 252 19.67 20.76 22.45 24.89 27.61 30.19 31.50 2001 248 21.80 22.37 23.85 26.24 30.64 34.20 36.34 2002 250 19.79 20.84 22.44 29.19 35.31 41.25 43.89 2003 252 16.45 16.78 18.96 21.21 26.92 34.77 35.80 8
2004 252 12.63 13.05 14.28 15.32 16.55 18.13 18.91 2005 252 10.75 11.08 11.66 12.52 13.64 14.83 15.58 2006 251 10.52 10.78 11.35 12.00 13.60 16.18 17.73 2007 251 10.34 10.97 13.11 16.33 21.65 25.24 26.48 2008 212 18.16 19.45 21.14 23.79 27.55 45.24 63.31 An imporan way of judging marke anxiey is o examine he persisence wih which VIX remains above cerain exraordinary levels. From Table 1, we know ha he chance of observing a VIX level above 34.22 is 5%. Suppose we re-examine he VIX hisory o coun he number of consecuive days ha VIX has remained above a level of 34.22. Four periods las more han 20 days can be idenified: Ocober 16 hrough December 22, 1987 (47 days), Augus 28 hrough Ocober 31, 2002 (46 days), Sepember 26 hrough Ocober 31, 2008 (26 days so far), and January 8 hrough February 8, 1988 (22 days). So, yes, we are experiencing abnormal behavior, bu, no, i is no unprecedened. We jus end o forge. VI. VIX predicion abou fuure volailiy Unlike he DJIA, which only has meaning relaive o is hisory, he VIX also has a simple, probabilisic inerpreaion concerning he expeced range of he rae of reurn on he S&P 500 index level over he nex 30 days. Figure 2 shown below provides a quick-and-ready mehod for inerpreing he level of VIX. 10 To undersand how o use he figure, assume ha he curren level of VIX is, say, 60. Reading up from 60 on he horizonal axis o he line labeled 50% and hen across o he verical axis, we see ha he expeced range of S&P 500 reurns over he nex 30 days is abou 11.5%. This means ha, if VIX is a 60, chances are 50-50 ha he rae of reurn on he S&P 500 index will be up or down by less han or more han 11.5% over he nex 30 days. The lines labeled 75% and 95% offer differen levels of probabiliy for a given VIX level. A a VIX level of 60, he 75% (95%) line indicaes ha expeced range of S&P 500 reurns over he nex 30 days is abou 20% (34%). In oher words, if he VIX is a 60, he chances ha he S&P will go up or down by less han 20% (34%) over he nex 30 days are 75% (95%). Conversely, a a VIX of 60, he chances ha he S&P will go up or down by more han 20% (34%) over he nex 30 days are 25% (5%). The figure is inended o be a quick reference 10 The figure makes wo simplifying assumpions: (a) he rae of reurn on he S&P 500 over he nex 30 days is normally disribued, and (b) he expeced rae of reurn on he S&P 500 over he nex 30 days is zero. Neiher assumpion is unreasonable. 9
guide for inerpreing VIX. For hose who wan o be more precise, he lines are generaed using he following relaions: 11 Expeced range a 50% = 0.1947 VIX Expeced range a 75% = 0.3321 VIX Expeced range a 95% = 0.5658 VIX A a VIX level of 60, he exac values for he expeced ranges are 11.68%, 19.92%, and 33.95%, respecively. Figure 2: Expeced range of S&P 500 reurns over he nex 30 days condiional on curren VIX level. Expeced range of S&P 500 reurns over nex 30 days 60 50 40 30 20 10 0 95% 75% 50% 0 20 40 60 80 100 VIX VII. VIX performance as a predicor An obvious quesion o ask abou he predicion rule described in he las secion is How well does i perform? To answer his quesion, a simple experimen was performed. A he beginning of each of he 274 monh in he sample period, he level of VIX was recorded. Based 11 These relaions are derived from he cumulaive sandard normal densiy funcion. A random number drawn from a uni normal disribuion has a 50% chance of being wihin 0.6745 sandard deviaions of 0, a 75% chance of being wihin 1.1504 sandard deviaions of 0, and a 95% chance of being wihin 1.9600 sandard deviaions from 0. Since VIX is an annualized sandard deviaion, we mus scale each of he coefficiens by he square roo of 12 o conver hem o monhly volailiies (e.g., 0.6745/ 12 =0.1947 ). 10
on he level of VIX a he beginning of he monh, 50%, 75%, and 95% expeced ranges were compued using he above formulas. The rae of reurn of he S&P 500 over he monh was hen compued, and he number of imes he reurn fell ouside he range during he 274 monhs of he sample period was recorded. Of he 274 rials, 95 or 34.7% fell ouside he 50% range, 20 or 7.3% fell ouside he 75% range, and 3 or 1.1% fell ouside he 95% range. In oher words, VIX works reasonably well as a predicor of he expeced of sock index movemens. 12 VIII. Creaion of oher volailiy indexes VIX is no unique as a sock marke volailiy index. I is merely he firs o have been inroduced and has a firs-mover advanage. The CBOE (2003) mehodology for compuing he index is no unique o he prices of S&P 500 index opions. I can be applied o any index opion marke. Indeed, he CBOE has already applied he mehodology o creae a volailiy index for he NASDAQ 100 (he "VXN") and for he DJIA (he "VXD"). The only imporan requisie is ha he underlying index opion marke has deep and acive rading across a broad range of exercise prices. No surprisingly, VIX has also araced imiaors inernaionally. The NYSE Euronex has applied he CBOE (2003) mehodology o index opions lised on he AEX (an index of 25 socks raded in Amserdam), he BEL20 (an index of 20 Belgium socks), he CAC40 (an index of 40 French socks), and he FTSE 100 (an index of 100 socks raded in he Unied Kingdom). Examining he co-movemens of he volailiy indexes in differen counries will undoubedly be a subjec of fuure research and discussion. IX. Summary VIX is a forward-looking index of he expeced reurn volailiy of he S&P 500 index over he nex 30 days. I is implied from he prices of S&P 500 index opions, which are predominanly used by he marke as a means of insuring he value of heir sock porfolios. While he curren levels of VIX are high by hisorical sandards, hey are no he highes hey have ever been nor have hey been as persisenly high during a leas wo episodes previously. 12 If anyhing, he range appears o wide relaive given he empirical resuls. One possible explanaion for his resul is ha invesors pay more han he acuarial value for heir porfolio insurance. 11
References Bollen, Nicolas P. B. and Whaley, Rober E., 2004, Does ne buying pressure affec he shape of implied volailiy funcions?, Journal of Finance 59, 711-754. Carr, Peer, and Madan, Dileep, 1998, Towards a heory of volailiy rading, in Risk Book on Volailiy. ed. by Rober A. Jarrow, Risk, New York, pp. 417-427. Chicago Board Opions Exchange, 2003. VIX: CBOE volailiy index. Working paper, Chicago. Demeerfi, Kresimir, Derman, Emanuel, Kamal, Michael, and Zou, Joseph, 1999, A guide o volailiy swaps, Journal of Derivaives 7, 9-32. Whaley, Rober E., 2006. Derivaives: Markes, Valuaion, and Risk Managemen. Firs ediion. (Hoboken, New Jersey: John Wiley & Sons, Inc.). Whaley, Rober E., 1993, Derivaives on marke volailiy: Hedging ools long overdue, Journal of Derivaives 1, 71-84. 12