HEDGING VOLAILIY RISK Menachem Brenner Sern School of Business New York Universiy New York, NY 00, U.S.A. Email: mbrenner@sern.nyu.edu el: 998 033 Fax: 995 473 Ernes Y. Ou Archeus Capial Managemen New York, NY 007 Email: ernie@archeuscap.com Jin E. Zhang School of Business and School of Economics and Finance he Universiy of Hong Kong Pokfulam Road, Hong Kong Email: jinzhang@hku.hk December 004 Keywords: Sraddle; Compound Opions; Sochasic Volailiy JEL classificaion: G, G3 Acknowledgemen: he pracical idea of using a sraddle as he underlying raher han a volailiy index was firs suggesed by Gary Gasineau in a discussion wih one of he auhors M. Brenner. We undersand ha Gary has applied for a U.S. paen on his volailiy insrumen, which is based on he same idea, he process used o implemen i and relaed insrumens. We would like o hank Seve Brown, Gary Gasineau, David Hai, Nikolay Halov, Pascal Nguyen, Mari Subrahmanyam, Alex Shapiro, Raghu Sundaram, Bruce uckman, David Weinbaum, and seminar paricipans a HKUS, Quaniaive Mehod in Finance 00 Conference, 00 APFA/PACAP/FMA Finance Conference for heir helpful commens. JEZ has been suppored by he Research Grans Council of Hong Kong under gran CERG-HKU 068/0H and he Universiy of Hong Kong.
HEDGING VOLAILIY RISK Absrac Volailiy risk plays an imporan role in he managemen of porfolios of derivaive asses as well as porfolios of basic asses. his risk is currenly managed by volailiy swaps or fuures. However, his risk could be managed more efficienly using opions on volailiy ha were proposed in he pas bu were never inroduced mainly due o he lack of a cos efficien radable underlying asse. he objecive of his paper is o inroduce a new volailiy insrumen, an opion on a sraddle, which can be used o hedge volailiy risk. he design and valuaion of such an insrumen are he basic ingrediens of a successful financial produc. In order o value hese opions, we combine he approaches of compound opions and sochasic volailiy. Our numerical resuls show ha he sraddle opion is a powerful insrumen o hedge volailiy risk. An addiional benefi of such an innovaion is ha i will provide a direc esimae of he marke price for volailiy risk.
I. Inroducion Risk managemen is concerned wih various aspecs of risk, in paricular, price risk and volailiy risk. While here are various efficien insrumens and sraegies o deal wih price risk, exhibied by he volailiy of asse prices, here is pracically only one ype of derivaive which deals wih volailiy risk, namely volailiy swaps, which is basically a forward/fuures conrac eiher on realized volailiy or on implied volailiy. In his paper we are inroducing a new volailiy insrumen, an opion on a forward-sar sraddle, which in our opinion dominaes he usefulness of exising alernaives, including volailiy swaps. While opion raders, in general, are subjec o volailiy risk, as well as oher risks, he main concern of dela-neural volailiy raders is he risk ha volailiy may change. hough many players may be being on he direcion ha volailiy may ake in he fuure and would no proec heir downside risk, some may seek o hedge heir bes a leas agains large movemens in volailiy. I is rue ha one can be on volailiy changes or hedge hem wih a sraegy ha combines holding of saic opions, all he ou-of-he money ones, and dynamically rade he underlying asse as shown by Carr and Madan 998. Such a sraegy, however, may be very cosly and no pracical for many users. Given he large and frequen shifs in volailiy in he recen pas 3 especially in periods like he summer of 97, he fall of 98 and he fall of 00, here is a growing need for insrumens o hedge volailiy risk. Pas proposals of such insrumens included fuures and opions on a volailiy index 4. he idea of developing a volailiy index was suggesed by Brenner and Galai 989 and 993 5. In 993 he Chicago Board Opions 3
Exchange CBOE has inroduced a volailiy index, named VIX, which was based on implied volailiies from opions on he SP00 index and is now based on he SP500 index and uses a differen mehodology 6. Only recenly did he Chicago Fuures exchange CFE inroduce a fuures conrac on he index. Opions have been planned for some ime now bu have no been inroduced ye. he main reason, in our opinion, ha i has aken so long o inroduce such derivaives is he lack of a cos-efficien radable underlying asse which marke makers could use o hedge heir posiions and o price hem. he firs heoreical paper 7 o value opions on a volailiy index is by Grunbichler and Longsaff 996. hey specify a mean revering square roo diffusion process for volailiy. heir framework is similar o ha of Hull and Whie 987, Sein and Sein 99 and ohers. Since volailiy is no rading hey assume ha he premium for volailiy risk is proporional o he level of volailiy. his approach is in he spiri of he equilibrium approach of Cox, Ingersoll and Ross 985 and Longsaff and Schwarz 99. A more recen paper by Deemple and Osakwe 000 also uses a general equilibrium framework o price European and American syle volailiy opions. hey emphasize he mean-revering in log volailiy model. Recenly, he CBOE has changed he mehodology ha was used o calculae VIX. he new forward-looking volailiy index uses curren opion prices o predic he nex 30 days realized volailiy. his volailiy index uses he S&P500 a-he-money pu and call opions and all ou-of-he-money opions weighed by he inverse of he square of heir srike prices. his approach is based on he work by Derman e al. 996 and Carr and Madan 998. he CBOE plans now o inroduce opions on he new VIX. he 4
exchange argues ha derivaives on his index will be an efficien ool o rade volailiy. Since, however, he underlying asse for hese derivaives is a combinaion of all available opions wih differen srikes i will no be a pracical replicaing porfolio, considering he low liquidiy of he ou-of-he-money opions and all he ransacions coss involved. Raher han an opion on an implied volailiy index or an opion on an index compued from he prices of many opions, some of which hardly rade, we presen here a proposal o inroduce an opion on a sraddle SO. he key feaure of he sraddle opion is ha he underlying asse is an A-he-Money-Forward AMF sraddle. he AMF sraddle is a raded asse priced in he marke place and well undersood by marke paricipans 8. Since i is AMF, is relaive value call + pu/sock is mainly affeced by volailiy. Changes in volailiy ranslae almos linearly ino changes in he value of he underlying, he AMF sraddle 9. hus opions on he AMF sraddle are opions on volailiy. We believe ha such an insrumen will be very aracive o marke paricipans, especially o marke makers. An addiional benefi of such an innovaion is ha i will provide a marke price for volailiy risk. A few recen papers have examined his issue empirically e.g., Coval and Shumway 00, Bakshi and Kapadia 003, Buraschi and Jackwerh 999. Examining differen sraegies hey conclude ha he volailiy risk premium is negaive. Currenly here is no marke ha calibraes his premium as, for example, he risk premium in he sock marke. he reurn on he underlying of he proposed produc, an always AMF sraddle, will provide such a calibraion. 5
In he nex secion we describe in deail he design of he insrumen. In secion III we derive he value of such an opion. Secion IV compares his insrumen o oher alernaives and secion V provides he conclusions. II. he Design of he Sraddle Opion One obvious class of users of hese opions are volailiy speculaors who buy and sell volailiy using sandard call and pu opions which are affeced by changes in he underlying asse and by ineres raes in addiion o changes in volailiy. hey simply may no be ineresed in such a package, especially if i coss more han a direc be on volailiy. he oher poenial classes of users are hedgers who mainly rade in he opions marke, like marke makers in opions, and porfolio managers who allocae funds beween socks and bonds using a mean-variance analysis. Since heir allocaion, and performance, may be affeced by an unexpeced change in volailiy hey may wan o insure agains volailiy risk. Again, his can be done using sandard sraddles bu his approach is inefficien since i insures agains boh: volailiy changes vega and changes in dela gamma. he price of he sraddle reflecs he broader coverage which is no sough afer. o isolae he volailiy risk one could dynamically rade he sraddle such ha i always is AMF bu such a sraegy enails ransacions coss ha become very high depending on he frequency of rebalancing which in urn depends on volailiy iself. hus, he desired insrumen, presened nex, should be a hedge agains volailiy risk only and should cos less han he alernaives including ransacions coss. 6
o manage he marke volailiy risk, say of he S&P500 index, a new insrumen, a sraddle opion or SO K,, SO wih he following specificaions, could be inroduced. A he mauriy dae of his conrac, he buyer has he opion o buy a hen a-he-money-forward sraddle wih a pre specified exercise price K SO. he buyer receives boh, a call and a pu, wih a srike price equal o he forward price, given he index level a ime. he sraddle maures a ime. he proposed conrac has wo main feaures: firs, he value of he conrac a mauriy depends on he volailiy expeced in he inerval o and herefore i is a ool o hedge fuure volailiy. I is sensiive o changes in volailiy bu no o changes in ineres raes or o large changes in he spo. Second, he asse underlying his opion is he conemporaneously raded sraddle 0. We believe ha, unlike he volailiy opions, his design will grealy enhance is accepance and use by he invesmen communiy. Compared o he available alernaives i is he mos cos effecive one see secion IV. he proposed insrumen is concepually relaed o wo known exoic opion conracs: compound opions and forward sar opions. Unlike he convenional compound opion his proposed opion is an opion on a sraddle wih a srike price, unknown a ime 0, o be se a ime o he forward value of he index level. In general, in valuing compound opions i is assumed ha volailiy is consan see, for example, Geske 979. Given ha he objecive of he insrumen presened here is o manage volailiy risk, we need o inroduce sochasic volailiy. 7
III. Valuaion of he Sraddle Opion In his secion we firs value he sraddle opion SO assuming deerminisic volailiy as our benchmark case. We hen apply a specific sochasic volailiy SV model o value he sraddle opion and illusrae is properies. A. he Case of Deerminisic Volailiy We firs analyze he case where volailiy changes only once and is known a ime zero. We assume a consan volailiy σ beween ime 0 and expiry dae of SO and a consan volailiy σ beween and mauriy dae of he sraddle S. Given is compound opion feaure, he derivaion of he value of SO a ime 0 involves four seps: Value of he underlying sraddle S a is mauriy, S, and nex a S. And hen payoff of SO a is expiry, SO, and finally a ime 0, 0 he payoff of sraddle S a is mauriy is: S r where K = S e and S = call + pu = S K S S is he sock price a., SO. Assuming ha he call and pu in he sraddle are European as is he ypical index opion and ha he Black-Scholes assumpions hold, and using he Brenner and Subrahmanyam 988 approximaion for AMF opions, we have S S α, π S = N d S σ where d = σ. he sraddle is pracically linear in volailiy. he relaive value of he sraddle, α = S / S is solely deermined by volailiy o expiraion. 8
he payoff of he sraddle opion SO a is expiraion is SO { S K,0} = { S } = max SO max α K SO, 3 hus he price of he SO a any ime, 0 < is, using he B-S model: SO r = α S N d K e N d σ, 4 SO where N x is cumulaive normal disribuion funcion and ln αs / K d = SO σ + r + σ he sensiiviy of SO o he volailiy in he firs period, called Vega, is. SO Vega = = S N' d, 5 σ where N ' d is he normal densiy funcion, which is a sandard resul for any opion excep ha d is also deermined by α which is in urn deermined by σ. hus, Vega in he firs period is affeced by volailiy in he second period which makes sense since he payoff a expiraion of SO is deermined by he volailiy in he subsequen period. is given by he sensiiviy of SO o he volailiy in he second period, called Vega, Vega SO = α = S N d = S N d N'. 6 σ d σ herefore, Vega is also a funcion of he volailiy in he curren period, no jus he volailiy of he subsequen period. Vega and Vega are proporional o he square roo of he lengh of each period. Each of hem depends on he volailiy in boh periods bu primarily on he volailiy in is own 9
period. hus, Vega could be smaller or larger han Vega depending on he volailiy in each period. B. he Case of Sochasic Volailiy We now urn o he case which is he very reason for offering a sraddle opion, he sochasic volailiy case. We assume a risk-neural diffusion process and a sochasic volailiy SV model similar o he one by Sein and Sein 99: ds = rs d + σ S db, 7 d σ = δ σ +. 8 θ d kdb Equaion 7 describes he dynamics of an equiy index S wih a sochasic volailiy σ. Equaion 8 describes he dynamics of volailiy iself which is revering o a long run mean θ where δ is he adjusmen rae and k is he volailiy of volailiy. wo independen Brownian moions. r is he risk-free rae. he condiional probabiliy densiy funcion of S is given by B and B are f S r r S, σ ; r,, δ, θ, k = e f S e 9 0 where f S 0 3/ S S = I η + cos η ln dη π S S 4 S in which he funcion I λ is given by equaion 8 of Sein and Sein 99. he ransiion probabiliy densiy funcion of σ is normal wih mean θ + σ θ e δ and variance k δ e / δ, 0
= exp,,, ; e k e e k k f δ δ δ δ θ σ θ σ δ π θ δ σ σ. 0 he join disribuion of S and σ is, f S f S f σ σ = since he wo Brownian moions are assumed independen. Using risk-neural valuaion wih he above join disribuion, he value of he sraddle S a ime is e S r r ds S S f e S S e S r =,,,, ; k r F S θ δ σ where he srike price S K is r S e. Given he values of he sraddle S, he price of he sraddle opion, SO, a ime = 0 can be compued as = 0 0 0 d f G SO σ σ σ σ 3 where 0 F K SO r ds S S f F K S e F G SO = σ σ σ σ 4 he values of SO are compued numerically in ables a o e using a range of parameer values. In choosing he parameer values we rely on he same empirical sudies ha Sein and Sein 99 do. In hese sudies, for example, he long-run volailiy is
beween 5 and 0 percen, he parameer k can have values as high as.5 and he mean reversion parameer can range from 4 o 0. Nex o he values from he SV model, in a, we presen he values using he BS model k=0. As expeced, he value of his compound opion using he SV model is larger han he value of his opion using he BS model. he difference beween he wo depends on he values of he oher parameers in he SV model and he srike price K. For relaively low srike prices, K SO, he effec of sochasic volailiy is raher small and he values are no ha differen from a BS value, ignoring sochasic volailiy. For higher srike prices ou of he money he effec of k, he volailiy of volailiy, is much SO larger. For K SO =, slighly ou-of-he-money, he value of SO a k=.3 is abou.6 imes larger han SO a k=.. vs. 0.47 while he BS value is only 0.36. able b shows he effec of iniial volailiy, σ 0. A low srike prices an increase in iniial volailiy has a small effec on he values of SO. A high srike prices he value of SO is lower bu he marginal effec of σ 0 is much higher. able c shows he effec of θ, he long-run volailiy on SO. For low values of θ, he value of SO is declining as we ge o he AM srike. able d shows he combined effec of volailiy and k, volailiy of volailiy, a he AM srike of SO. As expeced, he value of SO increases in boh and is raher monoonic. Sochasic volailiy has a relaively bigger effec in a low volailiy environmen. able e provides values of he sraddle opion for 3 mauriies of he sraddle. he values are higher for longer mauriies since he delivered sraddle has longer ime o expiraion and hus has a higher value. he effec is mos pronounced
when mauriy is one year. he SO has some posiive values even for srikes ha are way ou of he money. he effec of he various parameers on he value of SO could be discerned from he previous ables bu a beer undersanding of he complex relaionships can be obained from an examinaion of he various sensiiviies given in ables a o c. able a provides he sensiiviy of SO o changes in volailiy, which is he main issue here. able a provides hese values a 5 levels of σ 0. he values are high a all levels of iniial volailiy, hough hey end o decline as volailiy increases, indicaing ha changes in volailiy could be effecively hedged by he sraddle opion. I becomes less effecive as he srike price K SO increases, he opion is ou-of-he-money. able b provides values for he sensiiviy of SO o k, volailiy of volailiy. he higher is k, he higher is he vega of SO. I is mos sensiive a inermediae values of he srike price and approaches zero as he srike price increases. able c provides anoher ineresing sensiiviy, he sensiiviy wih respec o he ime o mauriy of he sraddle iself,. For a mauriy of 3 monhs he sensiiviy is higher han for a longer mauriy, 6 monhs or a year, because he incremenal value of SO a a shorer mauriy is larger han a a longer mauriy where he value is already high. An ineresing observaion regarding he value of SO emerges. Does SO have a higher value, relaive o BS value, in markes wih higher volailiy? I seems ha higher σ, for a given k volailiy of volailiy, ends o reduce he differences beween SV values and BS values since σ is he dominan facor in he valuaion. However, if higher σ is accompanied by higher k, SO values will be served lile by a sochasic volailiy model. hough he findings are an oucome of he model proposed by Sein & 3
Sein 99, some of he resuls are consisen wih general valuaion conceps. For example, AM opion valuaion, excluding jump processes, is largely dominaed by volailiy irrespecive of model specificaions. IV. he Sraddle Opion SO and Oher Alernaives he deerminans of an efficien volailiy-hedging insrumen are, is sensiiviy o volailiy, is radabiliy and is cos effeciveness. he key feaures of SO make i he mos efficien insrumen when compared o he alernaives. As discussed earlier, he AMF srike eliminaes he sensiiviy o ineres raes. he compound opion srucure which includes seing he AMF srike price a he delivery dae achieves wo hings; One, i provides an insrumen which is sensiive only o volailiy. wo, i provides an underlying asse which is radable. Oher alernaives are less efficien since hey respond o oher risks as well as volailiy and/or hey are more cosly. We examined he wo obvious alernaives; he curren AMF sraddle and he compound opion on a sraddle. We have simulaed he pahs of he index and volailiy and compued he coss of hese alernaives comparing hem o he one proposed here. We assume he equiy index S and is volailiy σ o follow he sochasic processes given in equaions 7 and 8. he iniial levels of S and σ are $00 and 0%, respecively. he parameers in he above processes are: r=3% dividend yield q = 0, δ = 8, θ = 0% and κ =0%. he oal period is year and we compare he values afer 6 monhs. We randomly drew 400 simulaion pahs for S and σ. Along each simulaed pah of equiy index S, we calculaed he corresponding values for sraddles and sraddle 4
opions a each fuure ime poin. he values from hese 400 pahs give us means and variances. he firs alernaive is an ourigh sraddle. We calculae he values of a year sraddle whose ime 0 srike price is se o be AMF. i.e. K=$03.05 given S = 00 and r = 3% a. he value of his sraddle, a ime 0, is $6.87 b. he value of he same sraddle afer 6 monhs, srike is sill $03.05 has a mean of 6.53 and a variance of 35.7. he mean a 6 monh ime is jus he value of he same sraddle a ime 0, plus he 3% carrying cos. he variance is large because he sraddle can be deep in or ou of he money a 6 monh ime, given is fixed srike level a $03.05. Now compare his o a sraddle whose srike price will be rese o he AMF level in 6 monhs ime. I.e., if S = 50 in 6 monh, hen he srike of his sraddle is se o be 5.7. he mean value of his sraddle is.7 and is variance is.75. his AMF sraddle has a much smaller mean and variance because is srike is rese o be a-hemoney-forward. I sill has some variance because he underlying S has a variance a 6 monh ime. We now compue he value of a sraddle opion call which a mauriy, in 6 monh ime, gives he holder he opion o buy a hen AMF sraddle. If he srike of his sraddle opion is.7, which is he mean value of he AMF sraddle, hen a ime 0 he value of his sraddle opion is 0.654 If he srike is 5, he value of he opion 6.6. So he oal cos of his alernaive; sraddle opion and he AMF sraddle is much less han he simple sraddle. 5
he second alernaive is a compound opion which a is mauriy a 6 monh ime, gives he holder he opion o buy a sraddle whose srike is fixed a $03.05 as in case. We know ha is mean value in 6 monh is $6.53. If he srike of his compound opion is 6.53, he mean value of is underling sraddle, hen he ime 0 value of his compound opion is.66 If he srike is 5, he value of he opion is.36; if he srike is.7, hen he value of he opion is 4.74. In oher words a he same srike price as he SO,.7, he price of he sraddle opion SO is much lower han he compound opion 0.654 vs. 4.74. Finally, in a coninuous and perfec marke one could dynamically replicae he sraddle opion by reseing he srike price wih every change in S bu in realiy even minimal ransacions coss will make such a proposiion very cosly. V. Conclusions he sochasic behavior of volailiy, which has always affeced opions premiums, has been, for he mos par, ignored by marke paricipans. However, any risk managemen sysem mus cope wih volailiy risk and i can do so in several ways; using exising insrumens, offered largely in he OC marke, and/or using a dynamic sraegy. Recenly, exchange raded fuures on volailiy have been added o he se of poenial insrumens. In his paper we presen a derivaive insrumen, an opion on a sraddle, ha can be used o hedge he risk inheren in sochasic volailiy. his opion could be raded on exchanges and used for risk managemen. As we show, i compares favorably wih oher possible alernaives; i is sensiive only o volailiy, he underlying asse is radable 6
and i is a cos effecive insrumen. Also, such an insrumen will provide a marke price for volailiy risk, which is currenly esimaed indirecly from exising insrumens. Since valuaion is an inegral par of using and rading such an opion we derive he value of such an opion using a sochasic volailiy model. We compare he value of such an opion o a benchmark value given by he BS model. We find ha he value of such an opion is very sensiive o changes in volailiy and herefore canno be approximaed by he BS model. References [] Bakshi, G. and N. Kapadia, 003, Dela-Hedged Gains and he Negaive Marke Volailiy Risk Premium, Review of Financial Sudies, 6, 57-566. [] Brenner, M. and D. Galai, 989, New Financial Insrumens for Hedging Changes in Volailiy, Financial Analys Journal, July/Augus, 6-65. [3] Brenner, M. and D. Galai, 993, Hedging Volailiy in Foreign Currencies, Journal of Derivaives,, 53-59. [4] Brenner, M. and M. Subrahmanyam, 988, A Simple Formula o Compue he Implied Sandard Deviaion, Financial Analyss Journal, 80-8. [5] Buraschi, A. and J. Jackwerh, 00, he Price of a Smile: Hedging and Spanning in Opion Markes, Review of Financial Sudies, 4, 495-57. [6] Carr, P. and D. Madan, 998, owards a heory of Volailiy rading, in Volailiy: New Esimaion echniques for Pricing Derivaives, R. Jarrow edior, Risk Books, London, 47-47. [7] Coval, J and. Shumway, 00, Expeced Opion Reurns, Journal of Finance, 56,983-009 7
[8] Cox, J. and M. Rubinsein, 985, Appendix 8A: An Index of Opion Prices, Opions Markes, Prenice-Hall, New Jersey. [9] Cox, J.C., J.E. Ingersoll and S.A. Ross, 985, A heory of he erm Srucure of Ineres Raes, Economerica, 53, 385-408. [0] Demeerfi, K., E. Derman, M. Kamal and J.Zou, 999, A Guide o Volailiy and Variance Swaps, Journal of Derivaives, 7, 9-3. [] Derman, E., M.Kamal, I.Kani and J. Zou, 996, Valuing Conracs wih Payoffs Based on Realized Volailiy, Goldman Sach & Co. manuscrip. [] Deemple, J. and C. Osakwe, 000, he Valuaion of Volailiy Opions, European Finance Review, 4, -50. [3] Galai, D., 979, A Proposal for Indexes for raded Call Opions, Journal of Finance, 34, 57-7. [4] Gasineau, G.L., 977, An Index of Lised Opion Premiums, Financial Analyss Journal, 34, 57-7. [5] Geske, R., 979, he Valuaion of Compound Opions, Journal of Financial Economics, 7, 63-8. [6] Grunbichler, A., and F. Longsaff, 996, Valuing Fuures and Opions on Volailiy, Journal of Banking and Finance, 0, 985-00. [7] Hull, J. and A. Whie, 987, he Pricing of Opions on Asses wih Sochasic Volailiies, Journal of Finance, 4, 8-300. [8] Longsaff, F.A. and E.S. Schwarz, 99, Ineres Rae volailiy and he erm Srucure, he Journal of Finance, 47, 59-8. [9] Lowensein, R., 000 When Genius Failed, Random House, New York. [0] Sein, E.M. and J.C. Sein, 99, Sock Price Disribuion wih Sochasic Volailiy: An Analyic Approach, Review of Financial Sudies, 4, 77-75. 8
Figure I S&P 500 Volailiy Index VIX 50.00 40.00 Volailiy 30.00 0.00 0.00 0.00 Jan-90 Sep-90 Jun-9 Mar-9 Nov-9 Aug-93 May-94 Jan-95 Oc-95 Jul-96 Mar-97 Dec-97 Sep-98 Jun-99 Feb-00 Nov-00 Aug-0 May-0 Jan-03 Oc-03 Jul-04 Dae Figure l Closing level on he S&P 500 Volailiy Index VIX. he sample period is January 3, 990 December 3, 004. Source: CBOE. 9
able a: he value of he Sraddle Opion, SO, a = 0 for a combinaion of srike price K SO and volailiy of volailiy k. S 0 =00, r = 0, σ 0, iniial volailiy,= 0.0, θ, long-run volailiy = 0.0, δ, reversion parameer = 4.00, =0.5, =.0. K SO k 0 BS 0.0 0.0 0.30 0.40 0.50 0.74.35.580.84.46.564 0.74 0.35 0.583 0.874.3.699 9.74 9.35 9.585 9.904 0.3 0.89 3 8.74 8.35 8.587 8.933 9.388 9.957 4 7.74 7.35 7.590 7.96 8.465 9.083 5 6.74 6.35 6.59 6.990 7.54 8.0 6 5.74 5.35 5.594 6.00 6.69 7.338 7 4.74 4.35 4.60 5.054 5.700 6.467 8 3.77 3.360 3.69 4. 4.793 5.60 9.308.408.73 3. 3.99 4.754 0.439.564.907.48 3.3 3.94 0.774 0.908.54.757.406 3.95 0.355 0.470 0.77.3.8.538 3 0.40 0.8 0.446 0.80.33.98 4 0.048 0.09 0.45 0.53 0.957.5 5 0.04 0.035 0.9 0.335 0.674.5 6 0.004 0.03 0.065 0.06 0.466 0.86 7 0.00 0.004 0.03 0.4 0.38 0.636 8 0.000 0.00 0.06 0.074 0.5 0.466 9 0.000 0.000 0.007 0.044 0.44 0.339 0 0.000 0.000 0.003 0.06 0.096 0.45 0
able b: he value of SO a = 0 for a combinaion of srike price KSO and he iniial volailiy σ 0. S 0 =00, r = 0, k = 0.0, θ = 0.0, δ = 4.00, =0.5, =.0. K SO σ 0 0.0 0.0 0.30 0.40 0.50 0.3.580.905.3.536 0.5 0.583 0.909.34.559 9.60 9.586 9.9 0.38 0.565 3 8.64 8.588 8.9 9.39 9.567 4 7.68 7.590 7.94 8.39 8.567 5 6.7 6.59 6.95 7.40 7.568 6 5.76 5.595 5.97 6.43 6.57 7 4.83 4.60 4.95 5.54 5.587 8 3.308 3.69 3.957 4.9 4.633 9.395.74 3.045 3.387 3.737 0.603.907.33.575.930 0.984.54.558.886.33 0.553 0.77.035.333.658 3 0.86 0.447 0.657 0.9.0 4 0.38 0.45 0.40 0.607 0.855 5 0.063 0.9 0.38 0.394 0.597 6 0.08 0.066 0.37 0.5 0.4 7 0.0 0.033 0.078 0.58 0.80 8 0.005 0.06 0.044 0.098 0.90 9 0.00 0.008 0.04 0.06 0.7 0 0.00 0.004 0.03 0.037 0.085
able c: he value of SO a = 0 for a combinaion of srike price K SO and he meanrevering level θ of volailiy. S 0 =00, r = 0, k = 0.0, σ 0 = 0., θ = 0.0, δ = 4.00, =0.5, =.0. K SO θ 0.0 0.0 0.30 0.4 0 6.53.533 6.566.933 5.573 0.568 5.676 0.954 4.67 9.58 4.733 9.965 3 3.680 8.587 3.759 8.970 4.733 7.590.769 7.973 5.86 6.59.773 6.973 6.06 5.595 0.773 5.973 7 0.548 4.60 9.773 4.973 8 0.49 3.69 8.774 3.973 9 0.00.74 7.776.974 0 0.036.907 6.783.974 0.0.54 5.803 0.977 0.004 0.77 4.853 9.983 3 0.00 0.447 3.953 8.999 4 0.000 0.45 3.30 8.09 5 0.000 0.9.406 7.084 6 0.000 0.066.796 6.75 7 0.000 0.033.305 5.35 8 0.000 0.06 0.95 4.56 9 0.000 0.008 0.64 3.787 0 0.000 0.004 0.437.36 able d: he value of SO a = 0 for a combinaion of he iniial volailiy σ 0 and he volailiy of volailiy k. S 0 =00, r = 0, θ = 0.0, δ = 4.00, =0.5, =.0, K SO =.5. σ 0 k 0.00 0.0 0.0 0.30 0.40 0.50 0.00 0.095 0. 0.548.03.633.389 0.0 0.73 0.408 0.745.30.850.6 0.0 0.535 0.663 0.99.474.095.854 0.30 0.849 0.965.77.748.364 3.7 0.40.94.99.59.048.654 3.397 0.50.558.654.930.369.96 3.69 0.60.937.06.85.707 3.84 4.000 0.70.35.408.654 3.059 3.6 4.3 0.80.7.800 3.034 3.43 3.969 4.654 0.90 3.7 3.00 3.43 3.798 4.38 4.997.00 3.537 3.606 3.88 4.8 4.695 5.349
able e: he value of SO a = 0 for a combinaion of srike price K SO and and differen mauriy spans of he sraddle. S 0 =00, r = 0, k = 0.0, σ 0 = 0.0, θ = 0.0, δ = 8.00, =0.5. 0.5 0.5.0 - K SO 0 8.07.466 6.73 7.089 0.487 5.96 6.093 9.494 4.308 3 5.093 8.497 3.34 4 4.093 7.497.36 5 3.095 6.497.37 6.9 5.497 0.37 7.54 4.497 9.37 8 0.6 3.503 8.37 9 0.57.534 7.37 0 0.09.668 6.37 0.08 0.975 5.30 0.008 0.505 4.334 3 0.00 0.35 3.378 4 0.000 0.00.49 5 0.000 0.040.74 6 0.000 0.05.4 7 0.000 0.005 0.673 8 0.000 0.00 0.38 9 0.000 0.000 0.06 0 0.000 0.000 0.07 3
able a: he sensiiviy of SO wih respec o σ 0. S 0 =00, r = 0, k = 0.0, θ = 0.0, δ = 4.00, = 0.5, =.0. 0.0 0.0 0.30 0.40 0.50 K SO σ 0 0 3.84 3.8 3. 3.4 3.5 3.4 3.7 3.6 3.5 3.6 3.7 3.5 3.6 3.7 3.9 3 3.3 3.4 3.5 3.7 3.30 4 3. 3.3 3.4 3.7 3.30 5 3.9 3. 3.4 3.6 3.9 6 3.7 3.0 3.3 3.7 3.3 7 3.6 3. 3.6 3.3 3.36 8 3.8 3.5 3.3 3.39 3.45 9 3. 3.6 3.37 3.46 3.55 0.94 3.8 3.36 3.49 3.6.54.90 3.8 3.40 3.55.03.45.83 3.4 3.37 3.38.89.36.76 3.07 4 0.87.34.84.30.68 5 0.50 0.89.35.83.7 6 0.7 0.55 0.94.39.85 7 0.3 0.3 0.63.0.47 8 0.6 0.9 0.4 0.74.4 9 0.03 0.0 0.6 0.5 0.87 0 0.0 0.05 0.6 0.36 0.65 4
FIGURE a he sensiiviy of SO o K SO a differen levels of iniial vol. 4.5 4 S e n s i i v i y 3.5 3.5 o f S O.5 σ0=0. σ0=0. σ0=0.3 σ0=0.4 σ0=0.5 0.5 0 0 3 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 K SO 5
able b: he sensiiviy of SO wih respec o k. S 0 =00, r = 0, σ 0 = 0.0, θ = 0.0, δ = 4.00, = 0.5, =.0. K SO k 0.0 0.0 0.30 0.40 0.50 0.6.78.8 3.5 4.85.6.93 3.9 4.04 5.30.6 3.04 3.74 4.57 5.77 3.6 3.6 4.7 5.09 6.5 4.6 3.7 4.58 5.60 6.7 5.6 3.38 5.0 6. 7.9 6.6 3.50 5.44 6.63 7.66 7.64 3.65 5.83 7.3 8.07 8.76 3.89 6. 7.55 8.55 9.0 4. 6. 7.76 8.87 0.50 4.46 6. 7.65 8.93.60 4.37 5.80 7.6 8.64.4 3.89 5.3 6.67 8.07 3.60 3.0 4.46 5.9 7.33 4 0.98.4 3.60 5.08 6.49 5 0.53.50.76 4. 5.6 6 0.6 0.94.0 3.38 4.77 7 0. 0.56.4.64 3.97 8 0.05 0.33 0.98.0 3.4 9 0.0 0.8 0.65.50.60 0 0.00 0.0 0.43.0.06 Figure b he Sensiiviy of SO Sensiiviy of SO 0 9 8 7 6 5 4 3 0 k=0. k=0. k=0.3 k=0.4 k=0.5 0 5 0 5 0 K SO 6
able c: he sensiiviy of SO wih respec o. S 0 =00, r = 0, k = 0., σ 0 = 0.0, θ = 0.0, δ = 8.00, - = 0.5. 0.5 0.50.00 K SO 0 0.00 0.000 0.000 0.00 0.000 0.000 0.00 0.000 0.000 3 0.003 0.000 0.000 4 0.003 0.000 0.000 5 0.003 0.000 0.00 6 0.003 0.00 0.004 7 0.004 0.007 0.08 8 0.08 0.045 0.093 9 0.4 0.7 0.07 0 0.394 0.38 0.336 0.708 0.559 0.44 0.74 0.585 0.444 3 0.5 0.47 0.40 4 0.58 0.33 0.33 5 0.05 0.80 0.39 6 0.037 0.094 0.65 7 0.0 0.046 0.09 8 0.003 0.0 0.069 9 0.00 0.009 0.043 0 0.000 0.004 0.05 Figure c Sensiiviy of SO Sensiiviy of SO 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 0 =0.5 =0.5 = 0 5 0 5 0 K SO 7
Endnoes A comprehensive analysis of volailiy and variance swaps is provided in Demeerfi, Derman, Kamal and Zou 999. One of he sraegies used by Long-erm-Capial-Managemen LCM was o sell volailiy on he S&P 500 index and oher European indices see Lowensein R. 000, p. 3. 3 he volailiy of volailiy can be observed from he behavior of a volailiy index, VIX based on he new mehodology, provided in Figure. 4 here were several aemps o inroduce volailiy derivaives e.g., he German DB launched a fuures conrac on he DAX volailiy index bu hose aemps were largely unsuccessful. 5 Gasineau 977 and Galai 979 have proposed an index of opion prices which corresponds o an implied volailiy index. Such an index is also described in Cox and Rubinsein 985. 6 he new mehodology which does no ransform he prices o implied volailiies provides a ime series which is very highly correlaed wih he old VIX series. 7 Brenner and Galai 993 use a binomial framework o value such opions where radabiliy is assumed implicily. 8 AMF sraddles are common in he FX marke and are quoed by implied volailiy. 8
9 Sricly speaking his is rue in a B-S world See Brenner and Subrahmanyam 988 bu here, wih sochasic volailiy, i may include oher parameers e.g. vol. of volailiy. 0 heoreically here is no difference if he delivered opion is a call, a pu or a sraddle since hey are all AMF. Pracically, however, here may be some difference in prices due, for example, o ransacions coss. A sraddle would provide a less biased hedge vehicle. Forward sar opions are paid for now bu sar a some ime in he fuure. A forward sar opion wih mauriy, can be regarded as a special case of he sraddle opion in which he srike price K SO is zero. If he volailiies and he periods happen o be he same hen he difference beween Vega and Vega will be negligible. 9