he Marke for olailiy rading; IX uures Menachem Brenner ern chool of Business New York Universiy New York, NY, U..A. Email: mbrenner@sern.nyu.edu el: 998 33, ax: 995 473 Jinghong hu chool of Inernaional rade and Economics Universiy of Inernaional Business and Economics Huixindongjie, Beijing, P. R. China Email: shujinghong@uibe.edu.cn Jin E. Zhang chool of Economics and inance and chool of Business he Universiy of Hong ong Pokfulam Road, Hong ong Email: jinzhang@hku.hk irs ersion: Augus 6 his ersion: May 7 ey words: olailiy rading; IX; IX uures JEL Classificaion Code: G3 * We would like o hank David Hai for his helpful commens. Jin E. Zhang has been suppored by a gran from he Research Grans Council of he Hong ong pecial Adminisraive Region, China Projec No. HU 747/6H.
he Marke for olailiy rading; IX uures Absrac his paper analyses he new marke for rading volailiy; IX fuures. We firs use marke daa o esablish he relaionship beween IX fuures prices and he index iself. We observe ha IX fuures and IX are highly correlaed; he erm srucure of IX fuures price is upward sloping while he erm srucure of IX fuures volailiy is downward sloping. o esablish a heoreical relaionship beween IX fuures and IX, we model he insananeous variance using a simple square roo mean-revering process. Using daily calibraed variance parameers and IX, he model gives good predicions of IX fuures prices. hese parameer esimaes could be used o price IX opions.
. Inroducion ochasic volailiy was ignored for many years by academics and praciioners. Changes in volailiy were usually assumed o be deerminisic e.g. Meron 973. he imporance of sochasic volailiy and is poenial effec on asse prices and hedging/invesmen decisions has been recognized afer he crash of 87. he indusry and academia have sared o examine i in he lae 8s, empirically as well as heoreically. he need o hedge poenial volailiy changes which would require a reference index has been firs presened by Brenner and Galai 989. In 993 he Chicago Board Opions Exchange CBOE has inroduced a volailiy index based on he prices of index opions. his was an implied volailiy index based on opion prices of he &P and i was raced back o 986. Unil abou 995 he index was no a good predicor of realized volailiy. ince hen is forecasing abiliy has improve markedly see Corrado and Miller 5 hough i is biased upwards. Alhough many marke paricipans considered he index o be a good predicor of shor erm volailiy, daily or even inraday, i ook many years for he marke o inroduce volailiy producs, saring wih over he couner producs like variance swaps. he firs exchange raded produc, IX fuures, was inroduced in March 4 followed by IX opions in ebruary 6. hese volailiy derivaives use he IX index as heir underlying. he curren IX is based on a differen mehodology han he previous IX, renamed XO, and uses he &P5 European syle opions raher han he &P American syle opions. Despie hese wo major differences he correlaion beween he levels of he wo indices is abou 98%. see Carr and Wu 6.
IX is compued from he opion quoes of all available calls and pus on he &P5 PX wih a non-zero bid price see he CBOE whie paper using following formula = i R σ e i, i i where he volailiy σ imes gives he value of he IX index level. is he 3 day volailiy esimae. In pracice opions wih 3-day mauriy migh no exis. hus, he variances of he wo near-erm opions, wih a leas 8 days lef o expiraion, are combined o obain he 3-day variance. is he implied forward index level derived from he neares o he money index opion prices by using pu-call pariy. i is he srike price of ih ou-ofmoney opions, i is he inerval beween wo srikes, is he firs srike ha is below he forward index level. R is he risk-free rae o expiraion. is he midpoin of he bid-ask spread of each opion wih srike i. i Carr and Madan 998, and Demeerfi e al 999 developed he original idea of replicaing realized variance by a porfolio of European opions. In epember 3, he CBOE used heir heory o design a new mehodology o compue IX. We now briefly review he heory behind equaion. If we assume ha he srike price is disribued coninuously from o becomes and neglec he discreizing error, equaion R R σ = e p d e c d ln. By consrucion, is very close o, hence is very small bu always posiive. Wih a aylor series expansion we obain he CBOE whie paper can be rerieved from hp://www.cboe.com/micro/vix/vixwhie.pdf 3
4 3 ln ln = = O. By omiing he hird order erms, 3 O, he las erm of equaion becomes ha of equaion. Carr and Madan 998 and Demeerfi e al 999 show ha due o he following mahemaical ideniy, =, max, max ln d d, he risk-neural expecaion of he log of he erminal sock price over srike is = ln R R d c e d p e E. Hence equaion can be wrien as, ln ln ln ln ln = = = = d E d d E E E σ σ where he las equal sign is due o Io s Lemma d d d ln σ =, under he assumpion ha he PX index follows a diffusion process, db d d σ µ = wih a general sochasic volailiy process, σ. o IX represens 3-day &P 5 variance swap rae. In pracice, he variance swap rae is quoed as volailiy insead of variance. I should be noed ha he realized variance can be replicaed by a porfolio of all ou-of-money calls and pus bu he IX index iself canno be replicaed by a porfolio of opions because he compuaion of he IX involves a square roo operaion agains he price of a porfolio of opions and he square roo funcion is nonlinear.
On March 6, 4, he newly creaed CBOE uures Exchange CE sared o rade an exchange lised volailiy produc; IX fuures, a fuures conrac wrien on he IX index. I is cash seled wih he IX. ince IX is no a raded asse, one canno replicae a IX fuures conrac using he IX and a risk free asse. hus a cos-of-carry relaionship beween IX fuures and IX canno be esablished. Our objecive is wo fold; irs, o use marke daa o analyse empirically he relaionship beween IX fuures prices and IX, he erm srucure of IX fuures prices and o esimae he volailiy of IX fuures prices. econd, o find parameer esimaes, using a simple sochasic volailiy model, ha bes describe he empirical relaionships and could possibly be used o price IX fuures and opions.. Daa In his paper, we use he daily IX index and IX fuures daa provided by he CBOE. he IX index daa, including open, high, low and close levels, are available from January 99 o he presen. he IX fuures daa, including open, high, low, close and sele prices, rading volume and open ineres, are available from March 6 4 o he presen. or each day we have four fuures conracs: wo near erm and wo addiional monhs on he ebruary quarerly cycle. or example, on he firs day of he lising, 6 March 4, four conracs 4, M4, 4 and X4 were raded which sand for he following fuures expiraion: May, June, Augus and November 4 respecively. he firs leer indicaes he expiraion monh followed by he expiraion year. he underlying value of he IX fuures conrac is IX imes under he symbol XB XB = IX. 3 5
he conrac size is $ imes XB. or example, wih a IX value of 7.33 on 6 March 4, he XB would be 73.3 and he conrac size would be $7,33. he selemen dae is usually he Wednesday prior o he hird riday of he expiraion monh. Our empirical sudy covers he period of wo years and eigh monhs from March 6 4 o November 6, wihin which here were 34 conrac monhs raded all ogeher. able provides a summary saisics of all of hem. he average open ineres for each conrac is 784, which corresponds o a marke value of 38 million dollars 3. he average daily rading volume for each conrac is 8, which corresponds o.5 million dollars. he shores conrac lased 35 days, while he longes 88 days. he average fuures price for each conrac changed from 85.6 for conracs ha maured in May 4 o 64. for conracs ha will maure in Augus 7 he average was aken for samples up o he mauriy dae or November 6, whichever is earlier, while he IX level ranged from 7.33 on March 6 4 o 9.9 on November 6. In general, he marke expeced fuure volailiy decreased during his period. 3. Empirical evidence 3.. he relaion beween IX fuures and IX XB Because he underlying variable of IX fuures, i.e. XB, is no a raded asse, we are no able o obain a simple cos-of-carry relaionship, arbirage free, beween he fuures price,, and is underlying, XB. ha is, r XBe, 3 Using he average IX fuures prices 35.5, we compue he marke value as 35.5 784 = 37,73,. 6
where r is he ineres rae, and is he mauriy. hus, we have gone o he daa o see wha we can learn abou he relaionship beween IX fuures prices and XB. We use his relaionship o esimae he parameers, in a sochasic volailiy model, ha could be used o price volailiy derivaives. here are four fuures conracs available on a ypical day. or example, on 6 March 4, we have four kind of IX fuures wih mauriies, in May, June, Augus and November, which corresponds o imes o mauriy of 53, 8, 4 and 3 days. We consruc 3, 6 and 9-day fuures prices by a linear inerpolaion echnique. or example, he 3-day fuures price is compued by using he marke daa of XB and May fuures on 6 March 4. he 6-day fuures price is compued by using he marke daa of May and June fuures. he 9-day fuures price is compued wih June and Augus fuures. We calculae hese fixed ime-o-mauriy fuures price on each day and obain hree ime series of 3-, 6-, and 9-day fuures prices. igure shows he ime series of XB and IX fuures for hree fixed ime-o-mauriies. Inuiively he four ime series are highly correlaed. able presens he correlaion marix beween he reurns of &P 5 index, XB and IX fuures. he reurn is compued as he logarihm of he price relaive on wo consecuive ends of day prices. All of he four series are negaively correlaed wih he &P 5 index. XB and IX fuures wih hree differen mauriies are almos perfecly correlaed. igure also shows ha he rading volume of IX fuures has been gradually increasing. igure shows he relaionship beween 3-day IX fuures and XB for he marke daa from 6 March 4 o November 6. In general, he IX fuures price is an increasing funcion of XB. he higher he IX XB, he higher is he price of IX fuures wih a given mauriy. 7
3.. he erm srucure of IX fuures price Over he period of March 6 4 o November 6, he average XB was 35.5. he average IX fuures prices were 44.8, 5.8 and 57.8 for 3-, 6- and 9-day mauriies respecively. he erm srucure of he average IX fuures price is upward sloping, which is demonsraed graphically in igure 3. he upward sloping IX fuures erm srucure indicaes ha he curren level of volailiy is relaively low compared wih he long-erm mean level and ha he volailiy is increasing o he long-erm high level. 3.3. he volailiy of XB and IX fuures Wih he ime series of XB and fixed mauriy IX fuures price, we compue he sandard deviaion of daily log price index relaives o obain esimaes of he volailiy of hese four series, assuming ha hese series follow a lognormal process. During he wo year period of our sudy we esimaed he volailiy of XB o be 84.4%, while he volailiies of IX fuures price are 35.7%, 3.% and 5.8% for 3, 6 and 9 day mauriies respecively. he longer he mauriy, he lower is he volailiy of volailiy. igure 4 shows he volailiy of XB and fixed mauriy IX fuures price. he erm srucure of IX fuures volailiy is downward sloping. he phenomenon of downward sloping IX fuures volailiy is consisen wih he mean-revering feaure of he volailiy. ince he long-erm volailiy approaches o a fixed level, long-enor IX fuures should be less volaile han shor-enor ones. 8
4. A heoreical Model of IX uures price 4.. IX fuures price We now use a simple heoreical model o price he fuures conracs using parameer esimaes obained from marke daa. We hen es he exen o which model prices can explain marke prices. In he physical measure, he PX index, sochasic variance model, is assumed o follow Heson 993 P d d db = µ, P P P d θ d σ db =, where µ is he expeced reurn, P θ is long-erm mean level of he insananeous variance, P is he mean-revering speed of he variance, σ measures he volailiy of variance, P db and P db are incremens of wo Brownian moions ha describe he random noises in PX index reurn and variance. hey are assumed o be correlaed wih a consan coefficien, ρ. By changing probabiliy measure from P o as follows, P µ r P λ db = db d, db = db d σ, where r is he risk-free rae, and λ is he marke price of variance risk, we obain he dynamics of he PX index in he risk-neural measure d rd db =, P P P d [ θ λ ] d σ db =, 9
where db and db are incremens of wo Brownian moions wih he correlaion ρ. We define risk-neural long-erm mean level θ and mean-revering speed as θ P θ, = P λ. λ = P P hen he risk-neural dynamics of he insananeous variance can be wrien as d = θ d σ db, 4 he ransiion probabiliy densiy, given by Cox, Ingersoll and Ross 985, is q / uv v f = ce I q u uv, 5 where c σ [ e =, ] u c e =, c θ v =, q =, σ and I is he modified Bessel funcion of he firs kind of order q. he disribuion q funcion is he noncenral chi-square, χ v;q,u, wih q degrees of freedom and parameer of noncenraliy u proporional o he curren variance,. Wih he risk-neural dynamics of he variance, we can evaluae he firs hree condiional momens of he fuure variance, s, < < s, as follows E E E e s = θ s θ, s s s e [ ] e s E s = σ e σ θ, 4 [ ] 4 3 s 3 σ σ θ s s e e s E s = e, 3 where E sands for he condiional expecaion in he risk-neural measure.
he IX index a curren ime is defined as he variance swap rae over he nex 3 calendar days. I is equal o he risk-neural expecaion of he fuure variance over he period of 3 days from o τ wih τ = 3/ 365, IX τ τ = E sds = τ τ τ = τ E ds s s [ θ θ e ] ds = B θ B, 6 where τ e IX B = is a number beween and. Hence τ is he weighed average beween long-erm mean level θ and insananeous variance wih B as he weigh. Noice ha he correlaion, ρ, does no ener ino he IX formula, hence he IX values do no capure he skewness of sock reurn. he price of IX fuures wih mauriy is hen deermined by [ B B ] = E XB = E IX = E θ = B θ B f d. 7 Equaions 6 and 7 deermine he IX fuures price,, as a funcion of is underlying, XB IX and ime o mauriy - wih hree parameers,, θ and σ as follows = XB, ;, θ, σ. 8 Given he values of parameers,, θ, σ and curren level of XB, we can back ou from equaion 6 and hen calculae wih equaion 7 he curren IX fuures price, for differen mauriies. Equaion 7 can be regarded as a closed-form formula for he IX fuures price.,
o furher he relaion beween and XB, we expand B B θ wih aylor expansion near he poin of E and obain [ ] [ ] [ ] [ ] / / / E B BE B BE B B B = θ θ θ [ ] [ ] 3/ 8 E B BE B θ [ ] [ ] [ ] 4 3 3 5 / 6 E O E B BE B θ, where O. sands for he higher order erms. aking expecaion in he risk-neural measure gives an approximae formula for he IX fuures price [ ] [ ] [ ] 3 6 8 6 3 3 5/ 4 3/ / O e e e B Be Be e e e B Be Be Be Be σ θ θ σ θ θ σ θ = 9 Numerical resuls show ha he approximae formula is very accurae for reasonable se of parameer values. Evaluaing IX fuures price wih he approximae formula is much faser. his proves o be very imporan in he model calibraion exercises when large numbers of price calculaion are required. 4.. Esimaing model parameers from IX values Given he physical process of he insananeous variance we can obain he likelihood funcion of he variance over one day. In equaion 6 we relae he IX index o he insananeous variance; hence we would be able o evaluae he likelihood of he IX over
he day. By maximizing he likelihood funcion over he las 4 rading days, we can P esimae he four parameers,, P θ, λ and σ for each day 4. he daily esimaed parameers are presened in igure 5. heir average values are 6.967,.49, -.46 and.35 respecively; heir sandard deviaions are.994,.98,.788 and.375 respecively. he negaive value of λ is consisen wih he empirical phenomenon of negaive volailiy/variance risk premium 5. We hen price IX fuures by using equaion 7 wih he esimaed parameers. he model price and marke price are presened in igure 6. In general, he model price seems o be highly correlaed wih he marke price, bu here is a non rivial gap beween he prices he roo mean squared error is abou -5% of he IX fuures price. 4.3. Calibraing he IX fuures price model We hen used he erm srucure of IX fuures o ge parameer esimaes ha may provide beer model prices. or example, on 6 March 4, he marke prices are given by XB =73.3,, = 3/365, 9.,, 3 = 6/365,.8 and 3, = mk 9/365,.5, we solve he following opimizaion problem min, θ, σ i= mk 3 i i mdl XB, i ;, θ, σ mk mk and obain he following values of he hree parameers 6 4 Zhang and Zhu 6 use he same mehod o esimae he parameers for he whole 99 o 5 period. 5 Coval and humway repor a negaive reurn of zero-bea, a-he-money sraddle posiions. Bakshi and apadia 3 examine he negaive marke volailiy risk premium by using Dela-hedged opion porfolios. Carr and Wu 3 sudy he variance risk premia on indexes and individual socks by using he reurn of variance swaps. 6 I akes 4 seconds for one calibraion exercise if we use he exac formula 7, however i akes only one second if we use he approximae formula 9. he resuls from wo formulas are very close each oher. or efficiency, we will use he approximae formula in our calibraion from now on. 3
= 7.646, θ =.4396, σ =.5. igure 7 shows he daily calibraed se of parameers. When we do he calibraion we se bounds for he hree parameers 4 8,. θ. 5, and. σ. 8. he calibraed values of and σ are somehow randomly disribued wihin he bounds. he calibraed values of θ seem o be more sable han he oher wo parameers. he average values of he hree parameers, θ and σ are 5.585,.359 and.5885 respecively. heir sandard deviaions are.546,.99 and.398 respecively. Wih he daily calibraed se of parameers we can recover he laen insananeous variance/volailiy variable by using formula 6. he resul is shown in igure 8 ogeher wih he IX index. he wo ime series seem o be highly correlaed. Wih he se of parameers and marke value of XB on day - we can compue he model price of IX fuures on he nex day,. We hen compare he marke prices wih he model prices. igure 9 shows ha he model prices are remarkably close o he marke prices of 3-, 6- and 9-day IX fuures. he roo of mean squared error beween he wo prices is around.3 dollars, which is less han % of he average IX fuures price of around 5. Our simple mean-revering variance model capures he dynamics of IX fuures price very well. 4.4. olailiy of he IX index and IX fuures rom equaions 6 and 4, applying Io s Lemma gives us a sochasic process for he IX index as follows dix IX = IX B θ 8 IX 4 B σ d IX Bσ db. 4
he volailiy of he IX index or XB is hen given by σ IX = Bσ IX. Wih a reasonable se of parameers, for example, = 8., σ =.35, IX = 3.55, =.489, we obain a volailiy of 85.%, which is close o ha compued from he marke values of he IX index. We now define volailiy hedge raio,, o be he sensiiviy of he IX fuures price wih respec o XB, i.e., = XB,. XB Wih he average XB = 35.5 and average values of he calibraed parameers = 5.585, θ =.359 and σ =.5885, we may obain as a funcion of mauriy. As shown in igure, he volailiy hedge raio is a decreasing funcion of ime o mauriy. I sars from for he XB, goes o zero for he IX fuures wih a very long mauriy. he volailiies of he IX fuures and XB are relaed by XB σ = σ XB. Because and XB are very close o each oher and < he volailiy of he IX fuures is smaller han he volailiy of XB IX. or example, if σ XB =.844, XB = 35.5, = 5.585, θ =.359 and σ =.5885, we may compue he volailiy of 3-day IX fuures price as follows 5
35.5 σ 3 =.53. 844 =.49, 44.8 35.5 σ 6 =.34. 844 =.8, 5.8 35.5 σ 9 =.89. 844 =.3, 57.8 which is close o he value,.357,.3 and.58 compued from he marke prices. Our simple model also explains he downward sloping erm srucure of he volailiy of IX fuures. 5. Conclusion Wih he enormous increase in derivaives rading and he focus on volailiy came he realizaion ha sochasic volailiy is an imporan risk facor affecing pricing and hedging. A new asse class, volailiy insrumens, is emerging and markes ha rade hese insrumens are creaed. he firs exchange raded insrumen is, IX fuures. I has been rading on he CBOE uures Exchange since 6 March 4. In his paper, we firs sudy he behavior of IX fuures prices using he marke daa from March 4 o November 6. We observe hree sylized facs:. he index, IX, and he 3 IX fuures prices are negaively correlaed wih he &P 5 index. IX and IX fuures wih hree differen mauriies are almos perfecly correlaed.. he erm srucure of he average IX fuures price is upward sloping. 3. he volailiy erm srucure of IX fuures is downward sloping. 6
he firs fac is no surprising. raders ofen ake long posiion in volailiy derivaives o hedge he risk in sock posiion. he second fac shows ha he long erm mean level of volailiy is higher han he curren level. he hird observaion is no well-known. In he second par of he paper we use a simple mean-revering variance model o esablish he heoreical relaionship beween IX fuures prices and is underlying spo index. Using he variance parameers calibraed from our model wih he marke daa a -, we can price IX fuures a ime condiional on IX a ime. An empirical sudy over he whole sample period shows ha our model provides prices ha are very close o he marke prices. he roo mean squared error beween he marke and model prices is around.3 dollars, which is less han % of he average IX fuures price of around 5. Our simple mean-revering variance model capures he dynamics of IX fuures price very well. References:. Bakshi, Gurdip, and Nikunji apadia, 3, Dela-hedged gains and he negaive marke volailiy risk premium, Review of inancial udies 6, 57-566.. Brenner, Menachem, and Dan Galai, 989, New inancial Insrumens for Hedging Changes in olailiy, inancial Analys Journal, July/Augus, 6-65. 3. Carr, Peer, and Dilip Madan, 998, owards a heory of volailiy rading. In Rober Jarrow Ed., olailiy esimaion echniques for pricing derivaives, London: Risk Books, pp. 47-47. 4. Carr, Peer, and Liuren Wu, 3, ariance risk premia, Working paper, Ciy Universiy of New York and Bloomberg L. P. 5. Carr, Peer, and Liuren Wu, 6, A ale of wo indices, Journal of Derivaives 3, 3-9. 7
6. Corrado, Charles J., and homas W. Miller, Jr., 5, he forecas qualiy of CBOE implied volailiy indexes, Journal of uures Markes 5, 339-373. 7. Coval, Joshua D., and yler humway,, Expeced opion reurns, Journal of inance 56, 983-9. 8. Cox, John C., Jonahan E. Ingersoll, Jr., and ephen A. Ross, 985, A heory of he erm srucure of ineres raes, Economerica 53, 385-47. 9. Demeerfi, resimir, Emanuel Derman, Michael amal, and Joseph Zou, 999, A guide o volailiy and variance swaps, Journal of Derivaives 6, 9-3.. Heson, even L., 993, A closed-form soluion for opions wih sochasic volailiy wih applicaions o bond and currency opions, Review of inancial udies 6, 37-343.. Meron, Rober C., 973, heory of Raional Opion Pricing. Bell Journal of Economics and Managemen cience 4, 4-83.. Zhang, Jin E., and Yingzi Zhu, 6, IX uures, Journal of uures Markes 6, 5-53. 8
able : ummary aisics for IX uures Conracs from 6 March 4 o November 6 Code Conrac No. of observaions Period covered IX uures prices Open ineres olume ar End Mean d Mean Mean 4 May 4 38 3/6/4 5/9/4 85.6 9. 66 48 M4 Jun 4 56 3/6/4 6/6/4 86.8 4.3 56 7 N4 July 4 35 5//4 7/4/4 63.5 3.9 96 86 4 Aug 4 3/6/4 8/8/4 93..3 697 35 U4 ep 4 4 7/9/4 9/5/4 78.6.8 533 58 4 Oc 4 37 8//4 /3/4 5.7 9. 6 4 X4 Nov 4 64 3/6/4 /7/4 9. 5.6 794 45 5 Jan 5 6 //4 /9/5 46..6 75 G5 eb 5 68 6/8/4 /6/5 7. 3. 374 4 H5 Mar 5 37 /4/5 3/6/5 7.4 7.9 8 54 5 May 5 68 9//4 5/8/5 56.5 7. 69 57 M5 June 5 6 3//5 6/5/5 4.9.4 78 38 5 Aug 5 86 /9/4 8/7/5 48.6 8.9 443 7 5 Oc 5 86 6//5 /9/5 43. 5.4 86 66 X5 Nov 5 88 //5 /6/5 5.6 8.3 333 9 Z5 Dec 5 4 //5 //5 3.7 3. 98 84 6 Jan 6 4 /8/5 /8/6..9 35 5 G6 eb 6 85 5/3/5 /5/6 5.7.6 346 46 H6 Mar 6 43 //6 3//6.8.6 677 64 J6 Apr 6 4 /7/6 4/9/6. 9.7 648 6 6 May 6 87 8/9/5 5/7/6 48.4 9. 567 8 M6 Jun 6 43 4//6 6/4/6 46. 3. 95 79 N6 Jul 6 4 5//6 7/9/6 54.5 7.5 93 74 6 Aug 6 64 //6 8/6/6 5.4 6.4 487 467 U6 ep 6 4 7/4/6 9//6 4.8.9 3788 4 6 Oc 6 43 8/8/6 /8/6 3.9 3.9 8 685 X6 Nov 6 78 3/8/6 /5/6 47.5 9.3 4957 64 Z6 Dec 6 45 9//6 //6 34. 5. 6849 48 7 Jan 7 4 //6 /7/7 4.7 7. 46 7 G7 eb 7 8 3/8/6 //7 56.7 5. 3839 9 H7 Mar 7 4 //6 3//7 36. 9.5 58 8 J7 Apr 7 4 //6 4/8/7 4.5 3.3 36 7 7 May 7 7 3//6 5/6/7 6.3 3.9 83 49 7 Aug 7 9 6//6 8/5/7 64. 4. 77 6 Average 5.5 784 8 Noe: he fuures conrac code is he expiraion monh code followed by a digi represening he expiraion year. he expiraion monh codes follow he convenion for all commodiies fuures, which is defined as follows: January-, ebruary-g, March-H, April-J, May-, June-M, July-N, Augus-, epember-u, Ocober-, November-X and December- Z. 9
able : he correlaion marix beween daily coninuously compounded reurns of &P 5 index, XB and fixed mauriy IX fuures compued based on he marke daa from 6 March 4 o November 6. he fixed mauriy IX fuures prices are consruced by using he marke daa of available conracs wih linear inerpolaion echnique. he daily coninuously compounded reurn is defined as he logarihm of he raio beween he price on nex day and he price on curren day. he upper riangle is he same as he lower riangle because he marix is symmeric. &P 5 index XB 3-day IX fuures 6-day IX fuures 9-day IX fuures &P 5 index XB -.7974 3-day IX fuures -.7748.84 6-day IX fuures -.687.698.84 9-day IX uures -.6975.675.875.849
3 5 rading olume XB 3-day IX uures 6-day IX uures 9-day IX uures 5 5 4-3-6 4-5-6 4-7-6 4-9-6 4--6 5--6 5-3-6 5-5-6 5-7-6 5-9-6 5--6 6--6 6-3-6 6-5-6 6-7-6 6-9-6 igure. XB and IX fuures price wih hree fixed ime-o-mauriies beween 6 March 4 and November 6. he XB ime series is from he CBOE. he fixed mauriy IX fuures prices are consruced by using he marke daa of available conracs wih linear inerpolaion echnique. he bar char shows he rading volume normalized by conracs of fuures of all mauriy on he day. 5 3-day IX uures 5 5 5 5 5 3 XB igure. he relaion beween 3-day IX fuures and XB March 6 4 November 6. he XB ime series is from he CBOE. he 3-day IX fuures prices are consruced by using he marke daa of available conracs wih linear inerpolaion echnique.
Average IX uures Price 6 55 5 45 4 35 3 5 9,57.8 6,5.8 3,44.8,35.5 3 6 9 ime o Mauriy igure 3. he erm srucure of average IX fuures price. he average IX fuures price is compued based on he fixed mauriy daa from 6 March 4 o November 6. he fixed mauriy IX fuures prices are consruced by using he marke daa of available conracs wih linear inerpolaion echnique. olailiy of IX uures Price.9.8.7.6.5.4.3....844.357.3 3 6 9 ime o Mauriy.58 igure 4. he annualized volailiy of IX fuures price. he annualized volailiy is compued based on he fixed mauriy daa from 6 March 4 o November 6. he fixed mauriy IX fuures prices are consruced by using he marke daa of available conracs wih linear inerpolaion echnique.
9 P 8 7 6 5 4 Jun4 Dec4 Jun5 Dec5 Jun6.8 θ P.6.4. Jun4 Dec4 Jun5 Dec5 Jun6 λ - - -3 Jun4 Dec4 Jun5 Dec5 Jun6 3
σ.8.6.4. Jun4 Dec4 Jun5 Dec5 Jun6 igure 5. he variance parameers esimaed from he hisorical IX index of he las 4 rading days. he average values of he esimaed four parameers P P, θ, λ and σ are 6.967,.49, -.46 and.35 respecively. heir sandard deviaions are.994,.98,.788 and.375 respecively. 4
3 - da y I Xuure spric ean dx B 5 5 5 Jun4 Dec4 Jun5 Dec5 Jun6 6 - da y I Xuure spric ean dx B 5 5 5 Jun4 Dec4 Jun5 Dec5 Jun6 9 - da y I Xuure spric ean dx B 5 5 5 Jun4 Dec4 Jun5 Dec5 Jun6 igure 6. A comparison of model prediced prices and marke prices of 3-, 6- and 9- day IX fuures. he upper solid line is he marke price. he dos are model prediced prices based on he parameers esimaed from he IX index of he las 4 rading days. he lower solid line is he XB value shifed downward by 9 for he purpose of an easy comparison. he average prices of 3-, 6- and 9-day IX fuures are 44.8, 5.8, 57.8. he roos of mean squared error beween model prediced price and marke price are 4.89,.97, 6.8. 5
9 8 7 6 5 4 Jun4 Dec4 Jun5 Dec5 Jun6.8.7 θ.6.5.4.3. Jun4 Dec4 Jun5 Dec5 Jun6 σ.8.6.4. Jun4 Dec4 Jun5 Dec5 Jun6 igure 7. he model parameer values calibraed from he marke prices of 3-, 6- and 9-day IX fuures. he average values of he hree parameers, θ and σ are 5.585,.359 and.5885 respecively. heir sandard deviaions are.546,.99 and.398 respecively. 6
.5. è!!!!.5..5 IX.9 Jun4 Dec4 Jun5 Dec5 Jun6 igure 8. he insananeous volailiy,, recovered from IX index and daily calibraed se of parameers. he upper line is he insananeous variance, he lower line is IX index divided by and shifed down by 9% for he purpose of an easy comparison. 7
3 - da y I Xuure san dx B 5 5 Jun4 Dec4 Jun5 Dec5 Jun6 6 - da y I Xuure san dx B 5 5 Jun4 Dec4 Jun5 Dec5 Jun6 9 - da y I Xuure san dx B 5 5 Jun4 Dec4 Jun5 Dec5 Jun6 igure 9. A comparison on model prediced prices and marke prices of 3-, 6- and 9-day IX fuures. he upper solid line is he marke price. he dos are model prediced prices based on he parameers calibraed from IX fuures prices on he previous day. he lower solid line is he XB value shifed downward by 9 for he purpose of an easy comparison. he average prices of 3-, 6- and 9-day IX fuures are 44.8, 5.8, 57.8. he roos of mean squared error beween model price and marke price are.56,.34,.7. 8
.8 Hedge rai o.6.4. 4 6 8 ime o mauriy HdayL igure. he volailiy hedge raio as a funcion of ime o mauriy. XB = 35.5, he se of parameers is aken o be he average calibraed values = 5.585, θ =.359 and σ =.5885. 9