HEDGING VOLAILIY RISK Menachem Brenner Sern School of Business New York Universiy New York, NY 00, U.S.A. Email: mbrenner@sern.nyu.edu Ernes Y. Ou ABN AMRO, Inc. Chicago, IL 60604, U.S.A. Email: Yi.Ou@abnamro.com Jin E. Zhang Deparmen of Economics and Finance Ciy Universiy of Hong Kong 83 a Chee Avenue Kowloon, Hong Kong Email: efjzhang@ciyu.edu.hk Firs version: Augus 000 his version: November 000 Keywords: Sraddle; Compound Opions, Sochasic Volailiy JEL classificaion :3 Acknowledgemen: he pracical idea of using a sraddle as he underlying raher han a volailiy index was firs raised by Gary Gasineau in a discussion wih one of he auhors M. Brenner. We would like o hank David Weinbaum for his helpful commens.
HEDGING VOLAILIY RISK Absrac Volailiy risk has played a major role in several financial debacles for example, Barings Bank, Long erm Capial Managemen. his risk could have been managed using opions on volailiy which were proposed in he pas bu were never offered for rading mainly due o he lack of a 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. Unlike he proposed volailiy index opion, he underlying of his proposed conrac is a raded ahe-money-forward sraddle, which should be more appealing o poenial paricipans. In order o value hese opions, we combine he approaches of compound opions and sochasic volailiy. We use he lognormal process for he underlying asse, he Orensein-Uhlenbeck process for volailiy, and assume ha he wo Brownian moions are independen. Our numerical resuls show ha he sraddle opion price is very sensiive o he changes in volailiy which means ha he proposed conrac is indeed a very powerful insrumen o hedge volailiy risk.
I. INRODUCION Risk managemen is concerned wih various aspecs of risk, in paricular, price risk and volailiy risk. While here are various insrumens and sraegies o deal wih price risk, exhibied by he volailiy of asse prices, here are pracically no insrumens o deal wih he risk ha volailiy iself may change. Volailiy risk has played a major role in several financial disasers in he pas 5 years. Long-erm-Capial-Managemen LCM is one such example, In early 998, Long-erm began o shor large amouns of equiy volailiy. Lowensein, R. 000 p.3. LCM was selling volailiy on he S&P500 index and oher European indexes, by selling opions sraddles on he index. hey were exposed o he risk ha volailiy, as refleced in opions premiums, will increase. hey did no hedge his risk. hough one can devise a dynamic sraegy using opions o deal wih volailiy risk such a sraegy may no be pracical for mos users. here were several aemps o inroduce insrumens ha can be used o hedge volailiy risk e.g., he German DB launched a fuures conrac on he DAX volailiy index bu hose were largely unsuccessful 3. Given he large and frequen shifs in volailiy in he recen pas 4 especially in periods like he summer of 97 and he fall of 98, here is a growing need for insrumens o hedge volailiy risk. Pas proposals of such insrumens included fuures and opions on a volailiy index. he idea of developing a volailiy index was firs suggesed by 3 4 he quoe and he informaion are aken from Roger Lowensein s book When Genius Failed 000, Ch.7. Anoher known example is he volailiy rading done by Nick Leeson in 93 and 94 in he Japanese marke. His exposure o volailiy risk was a major facor in he demise of Barings Bank see Gapper and Denon 996. Volailiy swaps have been rading for some ime on he OC marke bu we have no indicaion of heir success. he volailiy of volailiy can be observed from he behavior of a volailiy index, VIX, provided in Figure. 3
Brenner and Galai 989. In a follow-up paper, Brenner and Galai 993 have inroduced a volailiy index based on implied volailiies from a-he-money opions 5. In 993 he Chicago Board Opions Exchange CBOE has inroduced a volailiy index, named VIX, which is based on implied volailiies from opions on he SP00 index. So far here have been no opions offered on such an index. he main issue wih such derivaives is he lack of a radable underlying asse which marke makers could use o hedge heir posiions and o price hem. Since he underlying is no radable we canno replicae he opion payoffs and we canno use he no-arbirage argumen. he firs heoreical paper 6 o value opions on a volailiy index is by Grunbichler and Longsaff 996. hey specify a mean revering square roo diffusion process for volailiy similar o ha of 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 997 also uses a general equilibrium framework o price European and American syle volailiy opions. hey emphasize he mean-revering in log volailiy model. Since he payoffs of he opion proposed here can be replicaed by a selffinancing porfolio, consising of he underlying sraddle and borrowing, we value he opion using a no arbirage approach. he idea proposed and developed in his paper addresses boh relaed issues: hedging and pricing. he key feaure of he sraddle opion is ha he underlying asse is an a-he-money-forward AMF sraddle raher han a volailiy index. he AMF sraddle is a raded asse priced in he marke place and well 5 6 he same idea is also described in Whaley 993. Brenner and Galai 993 use a binomial framework o value such opions where radabiliy is implicily assumed. 4
undersood by marke paricipans. 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 7. hus opions on he AMF sraddle are opions on volailiy. We believe ha such an insrumen will be more aracive o marke paricipans, especially o marke makers. 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 provides he conclusions. II. he Design of he Sraddle Opion o manage he marke volailiy risk, say of he S&P500 index, we propose a new insrumen, a sraddle opion or SO K,, wih he following specificaions. A SO he mauriy dae of his conrac, he buyer has he opion o buy a hen a-he-money- forward sraddle wih a prespecified exercise price K. he buyer receives boh, a call SO and a pu, wih a srike price equal o he forward price, given he index level a ime 8. he sraddle maures a ime. Our 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 volailiy changes. Second, he underlying asse is a raded sraddle. We believe ha, unlike he volailiy opions, his design will grealy 7 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. 8 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 differences in prices due, for example, o ransacions coss. A sraddle would provide a less biased hedge vehicle. 5
enhance is accepance and use by he invesmen communiy. he proposed insrumen is concepually relaed o wo known exoic opion conracs: compound opions and forward sar opions 9. Unlike he convenional compound opion our 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 proposed here is o manage volailiy risk, we need o inroduce sochasic volailiy. III. Valuaion of he Sraddle Opion he valuaion of he sraddle opion SO will be done in wo sages. Firs, we value he compound opion on a sraddle assuming deerminisic volailiy as our benchmark case. In he second sage we use sochasic volailiy o value he opion and hen we relae he wo. A. he Case of Deerminisic Volailiy o ge a beer undersanding of he sochasic volailiy case we firs analyze he case where volailiy changes only once and is known a ime zero. We assume a consan volailiy beween ime 0 and expiraion dae of SO and a volailiy beween and expiraion dae of he sraddle S. 9 Forward sar opions are paid for now bu sar a some ime in he fuure. A forward sar opion wih mauriy, as our proposed sraddle, can be regarded as a special case of our sraddle opion in which he srike price K is zero. SO 6
We firs value he sraddle a, he day i is delivered. he sraddle has he following payoff a mauriy S =C +P max S S e,0 = ± r + max S e S,0 r wherec and P are he payoff of he call and pu respecively, S is he sock price a and S is he sock price a. Since he srike price is a-hemoney-forward a = r. K S e Assuming ha he opions are European as is he ypical index opion and ha he Black-Scholes assumpions hold we have S = C + P = S [ Nd ] S e [ Nd ] r where d ln S/ S r + = d = d S for he price of he sraddle a. In paricular for = we know ha See Brenner and Subrahmanyam 988 C = P / π * S 3 hus S S / π 4 7
he sraddle is pracically linear in volailiy. he relaive value of he sraddle, S / S is solely deermined by volailiy o expiraion. he value of he sraddle opion SO is he value of a compound opion where he payoff of SO a expiraion is max S K,0 SO = max[ αs K,0] SO 5 where α = 6 π Equivalenly, he payoff can be wrien as αmax [ S K / α,0] 7 SO hus he price of he sraddle, using he B-S model, a any ime, 0 < is SO = S Nd K e Nd 8 r α SO where d r ln αs/ KSOe + = 9 8
Equaion 8 gives he value of an opion on a sraddle 0 which will be delivered a ime. his is a compound opion ha is easy o value since he sraddle is a-he-money- forward on he delivery dae which reduces he valuaion o a univariae like case where he α erm includes he parameer. Using 8 and 9 we can derive all he sensiiviies of SO o changes in he various parameers. In paricular, we are ineresed in he sensiiviies of SO o he volailiy in he firs period, called vega, and in he second period, called vega. Vega is given by SO vega = = S N' d 0 where N 'd is he sandard 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, he volailiy ha will prevail in he second period. 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. his leads o he nex quesion; how does he change in affec SO? his is given by SO α vega = = SNd = S Nd N' d where d = and N ' d is he sandard normal densiy funcion a, he mauriy of SO. 0 I should be noed ha he value of SO is based on an approximaion o he value of he AMF sraddle, S. As argued before his is pracically indisinguishable from he heoreical value. 9
he sensiiviy of SO o he volailiy during he life of he sraddle iself is also a funcion of he volailiy in he curren period, no jus he volailiy of he subsequen period. Since his case is only our benchmark case, we have no derived he oher sensiiviies, like hea and gamma, ec. We would like now o urn o he case which is he very reason for offering a sraddle opion, he sochasic volailiy case. B. he Case of Sochasic Volailiy Several researchers have derived opion valuaion models assuming sochasic volailiy. We are deriving he value of a paricular compound opion, an opion on an AMF sraddle, assuming a diffusion process similar o he one offered by Hull and Whie 987, Sein and Sein 99, and ohers. We assume ha an equiy index, S, follows he process given by ds = rsd + SdB d = d + kdb 3 Where r is he riskless rae and is he volailiy of S. Equaion describes he dynamics of he index wih a sochasic volailiy. Equaion 3 describes he dynamics of volailiy iself which is revering o a long run mean where is he adjusmen rae and k is he volailiy of volailiy. B and B are wo independen 0
Brownian moions. o obain a valuaion formula for SO, he opion on a sraddle, we need o go hrough a few seps saring from he end payoffs values. Firs, o ge he index value and he volailiy a ime we inegrae equaions and 3. S S r d db = exp τ τ + τ τ = + e + k e db 5 τ he condiional probabiliy densiy funcion of S is given by where f S S, ; r,,,, k = e f Se 6 r r o S S f S I η η dη 3/ = cos ln o π S S + 4 S where he funcion I λ is given by equaion 8 of Sein and Sein 99. he condiional probabiliy densiy funcion of is given by f ;,,, k = e πk e e k e 7 since is normally disribued wih
mean E = + e and variance V k e = An example of he probabiliy densiy funcion of he volailiy,, is given in Figure. he volailiy is normally disribued wih a mean 0.. heoreically we can have a negaive value for volailiy bu pracically he probabiliy is less han 8 0 for reasonable parameer values. he join disribuion of S and is f S, = f S f 8 since he wo Brownian moions are independen. Once we have he join probabiliy densiy funcion, we can price any opions wrien on he asse price and/or he volailiy, including sraddle opions proposed in our paper. Since SO is a compound opion wrien on a sraddle, we have o evaluae he price of he sraddle a ime firs, hen use i as he payoff o evaluae he sraddle a ime 0. Using risk-neural valuaion he price of he AMF sraddle a ime is In he example in Figure we use he same parameer values ha are used by Sein and Sein 99.
r r = r S e S e S Se f S S ds = S F ;, r,,, k 9 r where he srike price is S e For he consan volailiy model where k and are zero he funcion F, derived in he las subsecion, can be approximaed by F = 0 A π which is almos idenical wih he Black-Scholes values, as menioned above equaion 3. In ables, ab and c we provide he values of he sraddle S compued from he sochasic volailiy SV model using various parameer values. volailiy, able and figure 3 provides he values of S for a combinaion of iniial and k volailiy of volailiy. he firs column provides sraddle values using he BS model wih a deerminisic volailiy k=0. As expeced, he value of S increases as for high levels of does and as k does. For low levels of. For example, when he effec of k is higher han is 0 percen and k is zero, i.e. volailiy is deerminisic, he value of S is 8.9 which will go up o 9.7 for k=. and.47 for k=.5. However, for of 50% he value of S a k=0 is 9 and i only goes o 0.0 for a high k=.5. In oher words, in a high volailiy environmen he marginal effec of k on 3
he value of a sraddle is raher small and he BS model provides values which are indisinguishable from a sochasic volailiy model. able ab shows he effecs of he mean reversion parameers. he higher is, he long-run mean, he higher is he value of S. he higher is he reversion parameer,, he lower is he value of he sraddle for high iniial, since i converges faser o he lower long-run mean. able c provides he values of S for differen mauriy spans of he sraddle. he value of he sraddle increases wih mauriy much more a lower iniial volailiy han a higher volailiy, which is expeced even when k=0. Sochasic volailiy does no change ha. Given he values of he sraddle we can now compue he value of he opion on he sraddle SO. he price of SO a ime =0 is given by where SO = G f d 0 0 0 K G F e S f S S ds r SO = K SO 0 F F he values of SO are compued numerically in able 3a o 3e using a range of parameer values. Nex o he values from he SV model, in 3a, we presen he values using he BS model k=0. As expeced, he value of his compound opion using he SV 4
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 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 is much larger. For K =, currenly approximaely a-he-money, he value of SO a k=.3 is abou 90 SO percen larger han SO a k=..75 vs. 0.9 while he BS value is only 0.77. able 3b shows he effec of iniial volailiy, 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 is much higher. able 3c shows he effec of, 0 he long-run volailiy on SO. For low values of, he value of SO is declining as we ge o he AM srike. Hedging agains changes in volailiy in a low volailiy environmen is no worh much. able 3d 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 enviromen. able 3e 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 which 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 4a o 4c. able 5
4a provides he sensiiviy of SO o changes in volailiy, which is he main issue here. able 4a provides hese values a 5 levels of. he values are high a all levels of 0 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 increases, he opion is ou-of-he-money. able 4b SO 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 4c 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, for a given k volailiy of volailiy, ends o reduce he differences beween SV values and BS values since is he dominan facor in he valuaion. However, if higher is accompanied by higher k SO values will be served lile by a sochasic volailiy model. IV. Conclusions As was eviden in several large financial debacles involving derivaive securiies, like Barings and LCM, he culpri was he sochasic behavior of volailiy which has affeced opions premiums enough o conribue o heir near demise. In his paper we 6
propose 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. 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. 7
References [] Brenner, M. and D. Galai, 989, New Financial Insrumens for Hedging Changes in Volailiy, Financial Analys Journal, July/Augus, 6-65. [] Brenner, M. and D. Galai, 993, Hedging Volailiy in Foreign Currencies, Journal of Derivaives,, 53-59. [3] Brenner, M. and M. Subrahmanyam, 988, A Simple Formula o Compue he Implied Sandard Deviaion, Financial Analyss Journal, 80-8. [4] Cox, J.C., J.E. Ingersoll and S.A. Ross, 985, A heory of he erm Srucure of Ineres Raes, Economerica, 53, 385-408. [5] Deemple, J. and C. Osakwe, 997, he Valuaion of Voilailiy Opions, Working paper, Boson Universiy. [6] Gapper, J. and N. Denon, 996, All ha Gliers; he Fall of Barings, Hamish Hamilon, London. [7] Geske, R., 979, he Valuaion of Compound Opions, Journal of Financial Economics, 7, 63-8. [8] Grunbichler, A., and F. Longsaff, 996, Valuing Fuures and Opions on Volailiy, Journal of Banking and Finance, 0, 985-00. [9] Hull, J. and A. Whie, 987, he Pricing of Opions on Asses wih Sochasic Volailiies, Journal of Finance, 4, 8-300. [0] Longsaff, F.A. and E.S. Schwarz, 99, Ineres Rae volailiy and he erm Srucure, he Journal of Finance, 47, 59-8. [] Lowensein, R., 000 When Genius Failed, Random House, New York. [] 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
[3] Whaley, R.E., 993, Derivaives on Marke Volailiy: Hedging ools Long Overdue, Journal of Derivaives,, 7-84 9
0 Appendix: Benchmark Values of S and SO Regarding he benchmark values of S and SO, we would like o se hem depending on a mean-revering deerminisic volailiy funcion, i.e., hen he average volailiy over he ime period beween and is given by And he average volailiy over he ime period beween [0, ] and is given by Especially for he case = 0, he average volailiy beween 0 and is he price of S a ime is given by equaion 4 and he price of SO a ime is given by equaion 8 wih and given by he formulas above. he firs columns of able, able 3a and able 3d are compued by using hese formulas., d d = e = +.. ] [ e e d e d + + = + = =. 0 0 e e + + =. ] [ e e d e d + + = + = =
Figure S&P 00 Volailiy Index VIX 50 40 High: 48.56 Low: 6.88 Average: 5.34 VIX 30 0 0 0 Apr-97 Oc-97 Apr-98 Oc-98 Apr-99 Oc-99 Apr-00 Oc-00 dae Figure. Closing level on he S&P 00 Volailiy Index VIX. he sample period is April, 997 November 3, 000. Source: CBOE.
Figure Probabiliy Densiy of Volailiy 0 8 6 4 0 0 0. 0. 0.3 0.4 Figure. An example of he probabiliy densiy funcion of he volailiy. he parameer values are he same as in Sein and Sein 99.
able he values of he sraddle S k 0 BS 0.0 0.0 0.30 0.40 0.50 0.00 6.9605 7.0657 7.490 8.093 9.086 0.845 0.0 8.9446 9.076 9.78 9.766 0.563.4783 0.0.744.3430.55.998.5078 3.88 0.30 3.7735 3.833 4.0098 4.39 4.7879 5.47 0.40 6.36 6.434 6.5679 6.8343 7.50 7.7497 0.50 9.004 9.0466 9.83 9.457 9.754 0.99 0.60.670.704.838.0369.338.734 0.70 4.3557 4.3908 4.5009 4.684 4.9478 5.938 0.80 7.0506 7.0746 7.8 7.3459 7.5834 7.8935 0.90 9.7494 9.764 9.8606 30.046 30.305 30.535.00 3.448 3.4499 3.546 3.6844 3.87 33.440 able : he values of he sraddle S for a combinaion of iniial volailiy volailiy of volailiy k. S = 00, = 0.0, = 4.00, - = 0.5 year. and Figure 3 he values of he sraddle S 35 30 5 0 5 0 5 0 k=0 k=0. k=0. k=0.3 k=0.4 k=0.5 0 0. 0.4 0.6 0.8 Figure 3: he values of he sraddle S for a combinaion of iniial volailiy volailiy of volailiy k. S = 00, = 0.0, = 4.00, - = 0.5 year. and 3
able ab: he values of he sraddle S for a combinaion of iniial volailiy and he mean-revering parameers, of volailiy. S = 00, k = 0., - = 0.5 year. = 0.0 = 0.0 = 0.30 = 0.40 = 0.0 = 0.0 = 0.0 = 4.00 = 4.00 = 4.00 = 4.00 = 4.00 = 8.00 = 6.0 0.00 4.748 7.490 0.709 4.0967 7.490 9.34 0.3086 0.0 6.357 9.78.5645 5.999 9.78 0.6 0.779 0.0 8.6366.55 4.767 8.0004.55.4747.4057 0.30.007 4.0098 7.0700 0.53 4.0098.987.637 0.40 3.8507 6.5679 9.599.589 6.5679 4.4909 3.099 0.50 6.547 9.83.0478 5.0447 9.83 6.53 3.9838 0.60 9.57.838 4.687 7.5357.838 7.8788 5.0083 0.70.9585 4.5009 7.09 30.073 4.5009 9.6506 6.0898 0.80 4.7000 7.8 9.7943 3.4909 7.8.4566 7.7 0.90 7.438 9.8606 3.3373 34.8785 9.8606 3.875 8.384.00 30.096 3.546 34.79 37.38 3.546 5.349 9.5760 able c: he values of S for a combinaion of iniial volailiy mauriy spans of he sraddle. S =00, k = 0.0, = 0.0, = 8.00. and differen - 0.5 0.5.0 0.00 5.47 9.34 4.76 0.0 6.4605 0.6 5.486 0.0 8.078.4747 6.945 0.30 9.836.987 7.3547 0.40.6538 4.4909 8.5667 0.50 3.50 6.53 9.908 0.60 5.4040 7.8788.3358 0.70 7.3080 9.6506.8498 0.80 9.5.4566 4.46 0.90.4 3.875 6.056.00 3.0644 5.349 7.79 4
able 3a: he value of SO a = 0 for a combinaion of srike price K SO and volailiy of volailiy k. S 0 =00, r = 0, 0 = 0.0, = 0.0, = 4.00, =0.5, =.0. K SO k 0 BS 0.0 0.0 0.30 0.40 0.50 0.744.35.580.84.468.5649 0.744 0.35 0.583 0.8747.37.6998 9.744 9.35 9.5857 9.9047 0.3 0.894 3 8.744 8.35 8.5879 8.9335 9.3885 9.9570 4 7.744 7.35 7.5900 7.96 8.4653 9.0839 5 6.744 6.35 6.59 6.9906 7.54 8.08 6 5.744 5.35 5.5949 6.000 6.696 7.338 7 4.745 4.357 4.608 5.0548 5.7004 6.4673 8 3.778 3.3607 3.69 4.0 4.7933 5.608 9.3080.4080.739 3. 3.996 4.7545 0.4398.5648.9074.483 3.37 3.944 0.7745 0.9086.54.7579.4064 3.956 0.3559 0.4700 0.778.34.8.5386 3 0.405 0.8 0.4468 0.803.338.98 4 0.0484 0.090 0.453 0.539 0.957.58 5 0.048 0.0358 0.90 0.335 0.6743.53 6 0.004 0.03 0.0657 0.063 0.4668 0.864 7 0.000 0.0045 0.038 0.48 0.386 0.6368 8 0.000 0.005 0.06 0.0746 0.5 0.4665 9 0.000 0.0005 0.0079 0.0444 0.44 0.339 0 0.0000 0.000 0.0039 0.063 0.096 0.454 5
able 3b: he value of SO a = 0 for a combinaion of srike price KSO and he iniial volailiy 0. S 0 =00, r = 0, k = 0.0, = 0.0, = 4.00, =0.5, =.0. K SO 0 0.0 0.0 0.30 0.40 0.50 0.36.580.9050.8.5358 0.53 0.583 0.909.339.5587 9.599 9.5857 9.9 0.375 0.5653 3 8.645 8.5879 8.94 9.386 9.5667 4 7.68 7.5900 7.936 8.39 8.567 5 6.70 6.59 6.948 7.400 7.5679 6 5.76 5.5949 5.970 6.45 6.575 7 4.83 4.608 4.953 5.537 5.5869 8 3.3084 3.69 3.9570 4.96 4.639 9.3948.739 3.0453 3.3869 3.737 0.607.9074.333.575.995 0.9836.54.5577.8859.38 0.557 0.778.0348.337.6578 3 0.860 0.4468 0.6574 0.98.03 4 0.378 0.453 0.408 0.6066 0.8545 5 0.068 0.90 0.379 0.394 0.5970 6 0.075 0.0657 0.374 0.55 0.44 7 0.08 0.038 0.0779 0.58 0.804 8 0.0050 0.06 0.0436 0.0983 0.895 9 0.00 0.0079 0.04 0.0607 0.73 0 0.0009 0.0039 0.034 0.0373 0.085 6
able 3c: he value of SO a = 0 for a combinaion of srike price K SO and he meanrevering level of volailiy. S 0 =00, r = 0, k = 0.0, 0 = 0., = 0.0, = 4.00, =0.5, =.0. K SO 0.0 0.0 0.30 0.4 0 6.57.533 6.566.9330 5.573 0.5675 5.6756 0.954 4.670 9.587 4.737 9.9653 3 3.6795 8.587 3.7590 8.9704 4.739 7.5900.7693 7.976 5.86 6.59.775 6.973 6.060 5.5949 0.773 5.9733 7 0.5478 4.608 9.7733 4.9733 8 0.488 3.69 8.7737 3.9734 9 0.00.739 7.7756.9735 0 0.036.9074 6.785.9743 0.09.54 5.803 0.9768 0.0036 0.778 4.855 9.9834 3 0.00 0.4468 3.958 8.9985 4 0.0004 0.453 3.96 8.088 5 0.000 0.90.4058 7.0837 6 0.0000 0.0657.7963 6.749 7 0.0000 0.038.3049 5.348 8 0.0000 0.06 0.946 4.555 9 0.0000 0.0079 0.64 3.7869 0 0.0000 0.0039 0.4366.360 able 3d: he value of SO a = 0 for a combinaion of he iniial volailiy 0 and he volailiy of volailiy k. S 0 =00, r = 0, = 0.0, = 4.00, =0.5, =.0, K SO =.5. 0 k 0.00 0.0 0.0 0.30 0.40 0.50 0.00 0.095 0.05 0.5477.08.6334.3890 0.0 0.79 0.408 0.7453.98.8503.606 0.0 0.5354 0.668 0.998.4736.0947.8540 0.30 0.8493 0.9649.77.7479.3637 3.69 0.40.939.988.59.0479.6535 3.3967 0.50.558.6543.999.3688.96 3.696 0.60.9365.055.85.7067 3.84 4.0003 0.70.35.408.6540 3.0590 3.607 4.34 0.80.74.8003 3.0340 3.433 3.9690 4.6538 0.90 3.67 3.00 3.49 3.7977 4.377 4.9966.00 3.537 3.6063 3.884 4.80 4.6947 5.3488 7
able 3e: 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, 0 = 0.0, = 0.0, = 8.00, =0.5. K SO - 0.5 0.5.0 0 8.070.4667 6.738 7.0894 0.487 5.968 6.093 9.4949 4.3088 3 5.0936 8.4970 3.343 4 4.0936 7.4973.364 5 3.095 6.4974.37 6.97 5.4974 0.37 7.54 4.4979 9.37 8 0.6 3.5038 8.373 9 0.575.5394 7.373 0 0.096.6684 6.378 0.089 0.9753 5.308 0.0084 0.5057 4.334 3 0.003 0.354 3.3787 4 0.0006 0.004.496 5 0.000 0.040.746 6 0.0000 0.053.43 7 0.0000 0.0057 0.6733 8 0.0000 0.00 0.388 9 0.0000 0.0008 0.067 0 0.0000 0.0003 0.07 8
able 4a: he sensiiviy of SO wih respec o 0. S 0 =00, r = 0, k = 0.0, = 0.0, = 4.00, = 0.5, =.0. 0.0 0.0 0.30 0.40 0.50 K SO 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 Figure 4a he Sensiiviy of SO Sensiiviy of SO 4.5 4 3.5 3.5.5 0.5 0 0=0. 0=0. 0=0.3 0=0.4 0=0.5 0 5 0 5 0 K SO 9
able 4b: he sensiiviy of SO wih respec o k. S 0 =00, r = 0, 0 = 0.0, = 0.0, = 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 4b 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 30
able 4c: he sensiiviy of SO wih respec o. S 0 =00, r = 0, k = 0., 0 = 0.0, = 0.0, = 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 4c 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 3