Pricing Futures and Futures Options with Basis Risk

Transcription

1 Pricing uures and uures Opions wih Basis Risk Chou-Wen ang Assisan professor in he Deparmen of inancial Managemen Naional Kaohsiung irs niversiy of cience & Technology Taiwan Ting-Yi Wu PhD candidae in he Deparmen of inancial Managemen Naional Kaohsiung irs niversiy of cience & Technology Taiwan and Insrucor in he Deparmen of Business Adminisraion Kao Yuan niversiy Taiwan Chien-Hua Li Maser in he Deparmen of inancial Managemen Naional Kaohsiung irs niversiy of cience & Technology Taiwan

2 Pricing uures and uures Opions wih Basis Risk Chou-Wen ang Ting-Yi Wu and Chien-Hua Li Absrac In his aricle assuming he sochasic behavior of basis as a modified Brownian bridge process we obain closed-form soluions of fuures and fuures opions generalizing he Black 976. The arrangemen permis he formulas of fuures and fuures opions o be funcions of spo price volailiy of spo reurn iniial basis basis volailiy as well as he correlaion coefficien beween basis and spo reurn. In he meanime i also ensures he basis o be zero a mauriy of fuures conrac. rom he numerical es he fuures call opion price is posiively relaed wih he correlaion coefficien beween basis and spo reurn bu is a negaively in he iniial basis value. Meanwhile he sign of correlaion coefficien deermines he relaionship beween he basis volailiy and he fuures call price. inally we empirically esed our model wih &P 500 fuures call opions daily daa. Comparing wih he Black 976 model our model ouperforms he Black s model due o eliminae sysemaic moneyness and ime-o-mauriy biases and has beer predicion power. In oal sample daa he mean errors in erms of index and percenage are and.0% for our model and -7.% for Black s model. Keyword uures uures Opions Basis risk Brownian bridge

3 I. Inroducion In recen years here has a seady growh in he number of financial asses which one migh properly call derivaive asses ha are available for rading on he organized exchanges. Among hese he mos noable are conracs of fuures and opions wrien on fuures conracs. In 999 oal rading volume of fuures and fuures opions on equiy and index was 48 million conracs in Chicago Mercanile Exchange. In 004 conrac volume reached 33 million conracs a year. This represens an average annual growh rae of 4% for he pas 5 years. This specacular growh promps many researchers o ake a close look a pricing beween he fuures conracs and he underlying index. Tradiionally he pricing of sock index fuures has been based upon he Cornell and rench 983 which is known as cos-of-carry model. Assuming ha markes are perfec Cornell and rench 983 derive he fuures prices for a sock or porfolio of socks wih consan dividend payou and ineres rae. They also exend heir model by inroducing he forward rae seasonal dividends and a simple ax srucure. Afer he prominen sudy of Cornell and rench 983 Ramaswamy and undaresan 985 provide closed-form soluions for opions on fuures conracs wih sochasic ineres rae model of Cox Ingersoll and Ross 985 hey also argue ha mispricing of opions on &P 500 fuures can beer explained by sochasic ineres rae models. Hemler and Longsaff 99 develop a general equilibrium model of sock index fuures prices wih sochasic ineres raes and marke volailiy. Their model allows he sock index fuures price o depend on he variance of reurn on he marke insead of jus he prices of raded asses. or commodiy fuures Gibson and chwarz 990 develop a wo-facor model where he firs facor is he spo price of he commodiy and he second facor is he insananeous convenience yield. chwarz 997 exends his model by inroducing

4 a hird sochasic facor he sochasic ineres rae. Hilliard and Reis 998 invesigae he pricing of commodiy fuures and fuures opions under he sochasic convenience yield sochasic ineres rae and jumps in he spo price. Neverheless all of heir models leave he marke price of convenience yield risk as a parameer in heir pricing formulas. Meanwhile sandard no-arbirage argumens leave no room for explici modeling of mean reversion via he drif of he spo commodiy price. Therefore by assuming normaliy of coninuously compound forward ineres raes and mean-reversion convenience yields and log-normaliy of he spo price of he underlying commodiy Milersen and chwarz 998 obain closed-form soluions for he pricing of opions on fuures prices as well as forward prices. Yan 00 proposes a general commodiy valuaion models for fuures and fuures opions o allow for sochasic volailiy and simulaneous jumps in he spo price and spo volailiy. They find ha he derived closed-form soluion of fuures price is no a funcion of eiher spo volailiy or jumps however numerical examples show ha in pricing opions on fuures. In sum he above aricles use he ax srucure convenience yield or dividend he erm srucure of ineres raes marke volailiy as well as jumps in he spo price and spo volailiy o indirecly model he difference beween log fuures price and log spo price he basis *. I is worh o noice ha we may no assure how many sae variables should be included in he basis funcion. In his aricle assuming he sochasic behavior of basis as a modified Brownian bridge process we obain closed-form soluions of fuures and fuures opions generalizing he Black 976. A Brownian bridge is a sochasic process ha is like a Brownian moion excep ha wih probabiliy one i reaches a specified poin a a specified ime. nder no arbirage assumpion he spo price and he fuures price * The definiion is used by Yan 00. 3

5 will converge a he expiraion of he fuures conrac. I means ha he basis value is zero. The modified Brownian bridge process ensures he basis o be zero a mauriy of fuures conrac. The seup allows for he prices of fuures and fuures opions o be funcions of spo price volailiy of spo reurn iniial basis basis volailiy as well as he correlaion coefficien beween basis and spo reurn. rom he numerical analysis he fuures call opion price is posiively relaed wih he correlaion coefficien beween basis and spo reurn bu is a negaively in he iniial basis value. Meanwhile he sign of correlaion coefficien deermines he relaionship beween he basis volailiy and he fuures call price. inally we empirically esed our model wih &P 500 fuures call opions daily daa. Comparing wih he Black 976 model he empirical es shows clear evidence supporing he occurrence of basis risk. The fuures opion model wih basis risk ouperforms he Black s model by producing smaller bias and beer goodness of fi. I no only eliminaes sysemaic moneyness and ime-o-mauriy biases produced by Black model bu also has beer predicion power. In oal sample daa he mean errors in erms of index and percenage are and.0% for our model and -7.% for Black s model. An ouline of his sudy is as follows: ecion II presens he model specificaion for fuures and fuures opions. ecion III shows he numerical analysis for he basis risk call opion model. ecion IV empirically ess he basis risk model. ecion V exends he model o discuss some Greek leers hedge sraegy and he fuures pu opions wih basis risk while ecion VI summarizes he paper. II. The valuaion framework In order o price fuures opions wih he basis risk he fuures formula should be remodeled. We assume ha he fuures price is affeced by boh of underlying asse 4

6 and basis risk which is a modified Brownian bridge process. Therefore deriving he price of basis risk is he firs sage and we remodel he fuures and fuures call opions laer. The process of underlying asse nder spo maringale measure we assume he underlying securiy follows a geomeric Brownian moion wih coninuous dividend-yield δ and consan insananeous drif r and volailiy. The process is: d = r δ d+ dw. W sands for an one-dimensional sandard Brownian moion defined on a filered probabiliy space Ω P. The fuures price wih basis risk. Defining he basis sing he definiion proposed by Yan 00 he basis is defined as from he log fuures price wih delivery dae subrac he log spo price. Tha is: = ln ln Boh and are considered in finance o be lognormal. Therefore he is a normal disribuion. 0 is he basis a ime zero wih mauriy a ime and an adaped random variable. In he absence of arbirage opporuniy he fuures prices will agree wih he spo price a mauriy. Tha is should be zero.. The basis follows modified Brownian bridge process We assume ha he basis in he aricle follows modified Brownian bridge process. A Brownian bridge is a sochasic process ha is like a Brownian moion excep ha wih probabiliy one i reaches a specified poin a a specified ime. This process 5

7 ensures ha he basis is known wih cerainy a he ime he fuures is iniiaed and a mauriy. Tha is he basis is known wih cerainy a he ime he fuures is iniiaed and approaches o zero a mauriy under no arbirage assumpion. Le he basis be a modified Brownian bridge in he ime inerval [0 ] wih 0= ln0-ln0 and = 0. The process is provided in Theorem I. Theorem I: nder spo maringale measure he basis process is as follows: d = d + ρ dw + ρ dw 3 where is he basis volailiy ρ is he correlaion coefficien beween he underlying securiy and he basis and dw is one-dimensional Brownian moion defined on he filered probabiliy space Ω P. We furhermore assume E dw dw = 0. Therefore we have E d d = ρd. The modified Brownian Bridge process considers he basis volailiy and allows for ineracions beween he underlying asse reurn and he basis. The soluion of 7 is as follows: 0 v = + ρ dws v + ρ dwz v 0 v 0<< 4 The proof of Theorem is shown in Appendix A. The basis is modeled direcly wihou knowing how many sae variables should be included. Moreover will approach o zero almos surely when =. 3. Pricing fuures wih basis risk Now we derive he closed-form soluion for fuures wih basis risk. In view of The feaure is verified by Klebaner 988 page 4. 6

8 he value of fuures is muliplies by exponenial of and displayed in Theorem II. Theorem II: The soluion for fuures price is as follows: + + = 0 exp W r s s δ 5 ] 0 [ Where is he fuures price a ime wih expiraion dae on he conrac. sing he echnique of momen generaing funcion he mean and variance of fuures value wih basis risk are provided as follows: [ ] + 0 exp 0 r E µ δ 6 [ ] [ ] + 0 exp 0 exp 0 r V µ δ 7 where dv v v v v + + = ρ µ 8 dv v v v v + + = 0 0 ρ 9 The proof of Theorem II is verified in Appendix B. The model has some imporan characerisics of observed fuures process. is he expecaion of fuures price is a funcion of spo price volailiy of spo reurn iniial basis basis volailiy as well as he correlaion coefficien of basis and spo reurn. However he fuures formula derived by Yan 00 is no a funcion of spo volailiy. econd he basis is modeled direcly wihou knowing how many sae 7

9 variables should be included. Third he basis is no only sochasic bu also a zero value a he mauriy of fuures conrac. I means ha spo price and he fuures price converge a he expiraion of fuures conrac. Maringale propery for fuures opion wih basis risk In his secion we verify ha a discouned call opion wih basis risk is a maringale under spo maringale measure. Assuming ha he call price wih basis risk is a funcion of and C = C one can creae a riskless hedge using he sandard echnique as oulined in Meron 973. We consruc a hedged porfolio by shor selling a basis risk call opion and buying n unis of fuures. The value of his porfolio V is Where e = dg = n C r δ G = C + n 0. By Io s lemma we obain + ρ + C C 0 d [ ] n C + ρ dw + ρ dw + Where is defined in equaion 9. Equaion is verified in Appendix C. When n = C he hedging porfolio is riskless. nder no arbirage assumpion he hedging porfolio can earn he riskless ineres rae. Therefore we have he following parial differenial equaion: 0 rc C r + C + C = Equaion is he same PDE as presened in Black and choles 973. By 8

10 eyman-kac formula we obain he call price on fuures a ime zero wih mauriy T under he spo maringale measure : Basis rt + C0 = e E T k 3 The price of fuures opion wih basis risk The call price wih basis risk is verified ha i has maringale propery in above secion. sing 3 and T in 5 we have fuures call price wih basis risk a ime zero and he model is presened in Theorem III. Theorem III: uures European call opion valuaion wih basis risk is as follows: C Basis 0 Where rt = 0 exp δ T + µ 0 T N d ke N d 0<T< 4 d 0 ln + r δ T + µ 0 T + 0 T = K 0 T 5 d 0 T d = 6 and N is he cumulaive probabiliy of sandard normal disribuion. The proof of Theorem III is shown in Appendix D. The basis risk model in 4 is somewha differen wih he model derived by he 9

11 Black 976. The basis risk model has he erm exp [ µ 0 ] T raher han exp rt in opions formula. This erm capures he basis risk in fuures conrac. III. Exension Taking he basis risk ino consideraion we re-proof he Greek leers of dela and gamma and fuures European pu opions model. Compuaion of Greek leers irsly we presen Dela and gamma for a fuures European call opion wih basis risk. The soluion is proposed in nex heorem. Theorem IV: Dela and Gamma of he basis risk call opion are as follows: C [ T + 0 T ] N d Dela = exp µ C Gamma = δ 7 [ δt + µ 0 T ] 0 π 0 T exp Where N and are defined in Theorem III. d e d The proofs are showed in Appendix E. rom equaion and 7 here is an implicaion for hedging raders. When n = C he hedging porfolio creaed in 0 is 8 riskless. Tha is he raders of he basis risk opion should buy C unis of fuures as defined in 7 o consruc a hedging porfolio o hedge he risk induced by one uni of basis risk opion. Comparing Dela in 7 and Black976 we know ha he hedger should buy more shares of fuures when we ake ino accoun he basis risk. Black 976 derived he fuures call opion as follows: C = exp rt 0 exp δ T N d kn 0 d 0

12 uures European pu opion wih basis risk We display he fuures European pu opion formula by pu-call pariy in his secion. irsly we derive he pu-call pariy under basis risk environmen. Then using he relaionship beween call and pu and call opions derived in heorem III we have he fuures European pu opion formula. The closed-form soluions of pu-call pariy and pu opions are presened in Theorem V. Theorem V: The pu-call pariy of basis risk opions is as follows: r T P + exp [ δ T + µ T ] = ke + C 9 The closed-form soluion of fuures European pu opion is as follows: P Basis 0 rt = ke N d 0 exp δ T + µ 0 T N d 0<T< 0 Where µ T and are defined in Theorem III. d The proof of Theorem V is shown in Appendix. d IV. Numerical analysis To invesigae he properies of fuures opion wih basis risk we show graphically numerical resuls in igure and. The opion prices are compued using 0 =00 K=95 r=0.03 δ =0.0 T=0.3 =0.5 =0.5 and subsiuing hese in equaion 4. To explore he effecs of iniial basis value 0 and correlaion beween basis and sock reurn ρ on fuures call opion price we le = 0.09 ρ range from - o + and 0 range from -0. o +0.. The resul is shown in igure. The soluion of call price in 4 is a plane ha increase wih respec o ρ bu decreases wih respec o 0. In words he call price is an increasing funcion of correlaion coefficien bu decreasing funcion of

13 iniial basis value. When he iniial basis value is less han zero he fuures price is less han spo price. Holding oher parameers unchanged including fuures price he iniial basis values change from < 0 o > 0 he spo prices decline. Therefore he call prices also decline. igure shows he impac of basis volailiy and correlaion beween basis and sock reurn on opion price. Le 0= 0. ρ range from - o + and range from 0 o +0.. In view of igure he soluion of opion price in 4 is a surface ha increase wih respec o ρ. However he sign of correlaion coefficien deermines he influence of he volailiy of basis on he value of call price. When ρ > 0 he value of call price wih basis risk is increasing wih he increase of he volailiy of basis. If ρ < 0 he value of call price wih basis risk is decreasing wih he increase of he volailiy of basis. igure The effec of iniial basis value and correlaion beween basis and sock reurn on fuures call opion price

14 igure The impac of basis volailiy and correlaion beween basis and sock reurn on fuures call opion price V. Empirical Tes We empirically es uures European call opion model wih basis risk modeled in secion III. As a comparison he Black model is used o be he compeing model in he empirical es. Empirical Daa The empirical es of he model is performed wih daily rading prices of &P 500 fuures opions a he Chicago Mercanile Exchange CME from January o June We chose he daa based on wo consideraions as Lim and Guo 000. irsly he opions have grea liquidiy in American. econdly &P 500 index &P 500 fuures and &P 500 fuures opions are used wildly in exising lieraure. Boh he underlying fuures and he opions on fuures expired on June

15 Opion prices are mached wih he neares corresponding fuures price preceding he opion ransacions. Boh he fuures and opions prices are quoed in index poins. The fuures prices in he sample period range from 39.8 o 9.8. The srike price of he opions ranged from 800 o 400. The wildly ranged srike price may cause large forecasing error bu ha acually responses he efficiency of models in all siuaions. There were 438 call opion prices in he sample. One common filering rule was applied o he raw sample before he daa were used in he empirical es. Tha is call prices ha are less han 0.5 are no used o miigae he impac of price discreeness he ick size for &P 500 fuures opion is Mos opion-pricing models assume coninuous price movemens while in real world he prices move in icks. Nandi 996 and Lim and Guo 000 use his rule wih heir &P 500 fuures opion daa. The filered sample consiss of 3999 call prices. A more-deailed profile of he raw daa is lised in Table. The dividend yield of &P 500 composie a 004 is.53% o.96% proposed by hlfelder 004Tripp 005 and Bary 005. We use.7% as he esimae of &P 500 s dividend yield in 005. The risk-free rae is calculaed from quoes of.. Table. Descripive aisics T 90 90> T 50 T <50 All Mauriy Moneyness /K Basiscall BLcall Basiscall BLcall Basiscall BLcall Basiscall BLcall <0.97 Number Mean D Number Mean D

16 Number Mean D Number Mean D Number Mean D All Moneyness Number Mean D The sample daa are daily rading prices of &P 500 fuures opions a he Chicago Mercanile Exchange CME from January o June There are divided ino 3 ime-o-mauriy and 5 moneyness groups so we have 4 groups alogeher. Treasury Bill prices. The.. 3-weeks Treasury Bills used in his sudy were aucioned by he Bureau of he Public Deb from Jan 005 o Jun The average of discouned quoes is used o calculae he yield. Parameer Esimaion To implemen opion-pricing models some unobservable parameers need o be esimaed using he observed-raded-opion prices in he sample period. or simpliciy we assume he parameers are consan. However as monhly esimaions grealy increased he fi we follow he spiri of many exising empirical aricles o use his mehod.or basis risk model four parameers of he process ρ and 0 need o be esimaed from he realized daa. Bu he is he only parameer needed ee Bakshi e al. 997 Chang e al. 998 and Lim & Guo

17 o be esimaed in Black-choles model. Tes of Model Performance The model s performance is examined using empirical es. We divided he ime-o-mauriy ino hree subgroups and he moneyness ino five. Including boh he suboals and grand oals here are 4 groups. We presen he Mean error Mean of Absolue Error MAE and Roo Mean quare Error RME saisics in erms of boh index and percenage o assess he efficiency of he wo compeing opions models. The index poin error is defined as he difference beween model price and acual price and he percenage error is he index poin error divided by he model price. Tha is as follows: e index = C C mo acual eindex e per = 00% C mo Where C is he model call opion price C is he acual call opion price mo acual eindex is he index poin error and e per is he percenage error. Table. Empirical Tes Resul T 90 90> T 50 T <50 All Mauriy Moneyness /K Basiscall BLcall Basiscall BLcall Basiscall BLcall Basiscall BLcall <0.97 Panel A: Index-Poin Error Mean MAE RME

18 Mean MAE RME Mean MAE RME Mean MAE RME Mean MAE RME All Moneyness Mean MAE RME Panel B: Percenage Error <0.97 Mean MAE RME Table II. Empirical Tes Resul coninued T 90 90> T 50 T <50 All Mauriy Moneyness /K Basiscall BLcall Basiscall BLcall Basiscall BLcall Basiscall BLcall Mean MAE RME Mean MAE RME

19 Mean MAE RME Mean MAE RME All Moneyness Mean MAE RME The empirical es resul is biseced ino wo panels. Panel A shows he index poins error in each subgroup as well as suboals and grand oals. Panel B presens he percenage error based on model price. Model prices are calculaed using he esimaed parameers. Those are figured ou from he sample daa in he same monh. The empirical es is o assess he performance of he model prices compared o acual prices using he parameer esimaed from he realized daa in he same monh. The ime-o-mauriy and he moneyness biases are examined o analyze he magniude of misspecificaion. Table presens he empirical resuls. The basis risk model shows beer good-of-fi wih no evidence of ime-o-mauriy or moneyness bias. In opposiion he Black s model demonsraes evidence of ime-o-mauriy and moneyness biases. Bias. The overall empirical performances show no significan bias for he basis risk model bu he Black s model has significan bias. The mean errors in index-poin erms and percenage erms in he sample for he basis risk model are small bu no for Black s model. or grand oals group hose are and.0% for he basis risk model and -7.% for Black s model. rom he angle of suboals and subgroups he basis risk model had no significan ime-o-mauriy and moneyness bias. However we noiced ha he Black s model presened sysemaic bias in boh erms of ime-o-mauriy and 8

20 moneyness. In every ime-o-mauriy group he Black model underprices shown by a negaive value he opion prices. The farher he mauriy is he severer he opions underprices. In every moneyness groups he Black s model also generally underpriced he opions. In index-poin erms he degree of underpricing is proporional o he moneyness. In percenage erm he degree of underpricing however is opposie proporional o moneyness. A few groups are excluded from he consisen resul. In he firs suboal group in Table II where /K< 0.97 he Black model ouperforms he basis risk model in 90> T 50 and T <50 subgroups. Bu boh models have large mean errors in /K< 0.97 suboal group. In paricular he Black s model has mean error of -80.5% where T 90 and he basis risk model has 3.3% where T <50. This may be due o deep-ou-of-money causing illiquidiy bias. We also found ha he relaive conen beween he MAE and is mean error in each subgroup of index-poin erms was larger for he basis risk model han he Black s model. or example in he seveneen subgroup in Table II Panel A where T 90 and /K.06 he Black s model generaes an MAE of and a Mean of while he basis risk model has 3.07 and 0.39 respecively. If we ake MAE/Mean as Lim and Guo 000 we obain.0 for Black s model and 8. for he basis risk model. This can be inerpreed as he evidence of he Black model s bias. When mos of he errors are of he same sign in each subgroups he mean error in index-poin erms will be close o is MAE. The basis risk model s prices are more disribued around he acual prices so he mean error is much smaller han MAE in index-poin erms. Goodness of i. The MAE of he Black model in grand oal group of index-poin erms is while i is for he basis risk model. In percenage erms he MAE is 36.4% for he Black model and 9.5% for he basis risk model. The 9

21 resuls in he RME saisics are consisen wih his. The basis risk model dominaes he Black s in mos of he groups. According o he empirical es we make a shor conclusion here. The smaller bias and he beer goodness of fi in he empirical es for he basis risk model over he Black s shows ha here is evidence of basis risk in he &P 500 fuures opions. The basis risk model is a beer specificaion han he model wihou basis risk. VI. Conclusion This sudy formulaes an alernaive fuures European opion model. This model differs from he Black 976 by assuming ha he fuure prices are influenced by he processes of underlying asse and basis risk. We furher assume he basis risk dynamics follow modified Brownian bridges which are like Brownian moions excep ha wih probabiliy one hese reach a specified poin a a specified ime. The fuures and fuures opions model wih basis risk has some feaures. irs he basis is modeled direcly wihou knowing he funcion form and how many sae variables should be included. econd he basis is no only sochasic bu also zero a he mauriy of fuures conrac. nlike Cornell and rench 985 he basis is consan or deerminisic. Lasly he fuures price is a funcion of spo price volailiy of spo reurn iniial basis basis volailiy as well as he correlaion coefficien of basis and spo reurn. The fuures model derived by Yan 00 is no a funcion of spo volailiy. Theoreically he opion model wih basis risk is superior o assuming ha he fuure prices follow sandard cos-of-carry model. Because i has many characerisics saed above uses more parameers and herefore allows for more degree of freedom. The numerical es shows ha he fuures call opion price is posiively relaed wih he correlaion coefficien beween basis and spo reurn bu is a negaively in he 0

22 iniial basis value. Meanwhile he sign of correlaion coefficien deermines he relaionship beween he basis volailiy and he fuures call price. This heoreical superioriy has been empirically esed by &P 500 fuures opions daily daa. Comparing wih he Black s model he empirical es shows clear evidence supporing he occurrence of basis risk in fuures opions on sock index. Our model ouperforms he Black s model by producing smaller bias and beer goodness of fi. I no only eliminaes sysemaic moneyness and ime-o-mauriy biases produced by Black model bu also has beer predicion power. In overall sample daa he mean errors in erms of index and percenage are and.0% for our model and -7.% for Black s model. Lasly we derive he dela gamma and fuures European pu opion and discuss he hedge sraegy while aking he basis risk ino consideraion. Appendix A The basis in 3 can be rewried as follow: d = d + dw * A- Where dw * = ρ dw + ρ dw We use he following echnique o derived he soluion of. The echnique is presened by Klebaner 998 a page. Consider general linear sochasic differenial equaion DE in one dimension d = α + β d + γ + δ dw * A- Where funcions α β γ δ are given adaped processes and are coninuous funcions

23 of. Then is found o be α u δ u γ u γ u * = X 0 + du + dw u 0 X u 0 X u Where X is sochasic exponenial DE s and as follow: dx * = β X d + δ X dw A-3 rom A- and A- we have α = 0 β = γ = δ = 0. rom previous condiion we use A-3 o obain Eq. 4. Appendix B rom we have = exp[] B- sing 4 and he soluion of he Eq. 5 is derived. By B- 0 can be obained and subsiued ino 5. The ransformed funcion for 5 urns ou o be: = exp r δ v dw v 0 B- Where v v ρ + u = ρ u v dw and dw v = dw v v According o B- and momen generaing funcion of lognormaliy we obain Eq. 6 and 7. Appendix C nder spo maringale measure he sock price follows equaion. Applying Ios lemma o 5 we ge:

24 d = r δ + ρ + d + + ρ dw ρ dw + C- By assuming he call opion wih basis risk is a funcion of and. Applied Ios lemma we obain + C + ρ + [ ] + C d dc = C r δ + ρ + ρ dw + ρ dw + C- rom 0 we know dg = dc + nd C-3 ubsiue C- and C- ino C-3 Equaion is compued. Appendix D Proof of Theorem III. ubsiue 5 ino 3 he fuures European call opions a ime zero are as follows: C 0 = e = e rt rt E E T k + rt T D ke E D D- We divide he RH of D- ino wo erms. or he firs erm I on he RH of D-: I T rt T 0 = e E 0 exp r δ T + v dw v 0 T 0 = 0 exp δ + T T 0 Where denoes he Euclidean norm in. v du E exp 0 0 T T v dw v v dv D D- Assume ha here exiss a unique spo maringale measure R on Ω which D 3

25 is given by he Radon-Nikodym derivaive dr T T ξ = = T exp v dw v v d v D-3 d 0 0 where R is he vecor of marke prices of risks corresponding o he sources of randomness in he economy. By Girsanov s heorem he process R W defined by dw R dw d = D-4 is a sandard Brownian moion under probabiliy measure R. Hence I nder R-measure = 0 exp δ T + 0 T ER [ T K] D-5 µ T R lnt= ln 0 + r δ + T + µ 0 T + v dw v D-6 Where µ 0 and are defined in 8 and 9. Nex ubsiue D-6 ino D-5 we have I = 0 exp δ T + µ 0 T N d D-7 Where d is defined in 5. 0 or he second erm I on he RH of D-: I = ke rt E T > k rt 0 T = Ke P ln 0 + r δ T + + T 0 v dw v ln K rt = ke N d D-8 4

26 Where d Theorem III is compleed. is defined in 6. ubsiue D-7 and D-8 ino D- The Eq.4 in Appendix E sing 4 and chain rule he dela is as follows: Dela [ δ T + 0 T ] N d + C = = exp µ [ ] d N rt N d 0 exp T + µ 0 T ke δ E- Where d N d N = d d = 0 π T e d According o 6 we have d = - d T rom previous derivaions he second erm on he RH of E- can be expressed as follow: 0 exp [ δ T + µ 0 T ] 0 π T e d - T = ke rt 0 π T e d E- We use a similar rick o derive he soluion of he hird erm on he RH of E-. The soluion is as follow: ke rt d N = ke rt 0 π T e d E-3 5

27 We found ha E- is jus he same wih E-3. Therefore he Eq. 7 in Theorem IV is obained. ince we have he soluion of dela we obain Gamma C = = exp exp = Dela = [ δ T + µ 0 T ] [ δ + µ 0 T ] 0 π T N d T e d Appendix Proof of Pu-Call pariy in Theorem V irsly we inroduce wo noaions: indicaor D sands for T K and indicaor E means for K T. nder spo maringale measure a fuures European pu opions price a ime is as follows: P PT = E f B ke T E T = E f BT k T + T D E BT = f Bk B T BCT E + f E f E BT BT BT = f = T exp[ δ T + µ T ] C r ke + Therefore he pu-call pariy in Eq.9 is obained. By he derived pu-call pariy a fuures pu opion is derived. Bibliography. Bary A. 005 Barron s Insigh: I May be Big ocks Turn To Bea mall 6

28 Ones a Las Wall ree Journal. Easern ediion. New York N.Y Jan 9 pp... Black. and choles M. 973 The Pricing of Opion and Corporae Liabiliies Journal Poliical Economy Black. 976 The Pricing of Commodiy Conracs Journal of inancial Economics 3 pp Cakici N. and Chaerjee. 99 "Pricing ock Index uures wih ochasic Ineres Raes" The Journal of uures Markes pp Cakici N. and hu J. 00 Pricing Eurodollar uures Opions wih he Heah-Jarrow-Moron model The Journal of uures Markes 3-7 pp Chang J. and Loo J. 987 Marking o Marke ochasic Iners Rae and Discouns on ock Index uures The Journal of uures Markes 7 pp.. 7. Chiang R. and Okunev J. 993 An Alernaive ormulaion on he Pricing of oreign Currency Opions The Journal of uures Markes Vol. 3 No.8 pp Klebaner ima 998 Inroducion o sochasic calculous wih applicaions Imperial college press 336 pp. 9. Clifford A.B. and Torous N. T. 986 uures Opions and he Volailiy of uures The Journal of inance pp Cornell B. and rench K. R. 983a The Pricing of ock Index uures The Journal of uures Markes 33- pp.-4.. Cornell B. and rench K. R. 983b Taxes and he Pricing of ock Index uures Journal of inance pp Dwyer G. P. Locke P. and Yu W. 996 Index Arbirage and Nonlinear Dynamics beween he &P 500 uures and Cash Review of inancial udies 9pp Gibson R. and chwarz E. 990 ochasic Convenience Yield and Pricing of Oil Coningen Claims Journal of inance 45 pp Hemler M. L. and Longsaff. A. 99 General Equilibrium ock Index uures Prices: Theory and Empirical Evidence Journal of inancial and uaniaive Analysis 63-3 pp Hilliard J. and Reis J. 998 Valuaion of commodiy fuures and opions under sochasic convenience yields ineres raes and jump diffusions in he spo Journal of inancial and uaniaive Analysis. eale: Mar 998.Vol.33 Iss. ; pg. 6 6 pgs 6. Klemkosky R. C. and Lee J. H. 99 Index-uures Arbirage and he Behavior of ock Index uures Prices Journal of uures Markes 7

29 pp Krishna R. and uresh M. 985 The Valuaion of Opions on uures Conracs The Journal of inance. No.5 pp Lim K.G. 990 Arbirage & Price Behavior of he Nikkei ock Index of uures Journal uures markes pp Lim K. G. and Guo X. 000 Pricing American Opions wih ochasic Volailiy: Evidence from &P 500 uures Opions Journal uures markes 0 pp Mackinlay A. C. and Ramaswamy K. 988 Index-uures Arbirage and he Behavior of ock Index uures Prices. Review of inancial udies pp Meron R. C. 973 "Theory of Raional Opion Pricing" Bell Journal of Economics & Managemen cience Vol. 4 pp Menachem B. and Mari. 985 Opions on he po and Opions on uures The Journal of inance pp Michael. and Muinul C. 999 A imple Non-parameric Approach o Bond uures Opion The Journal of ixed Income pp Miffre J. 004 The Condiional Price of Basis Risk: An Invesigaion sing oreign Exchange Insrumens Journal of Business inance & Accouning 3 pp Ramaswamy K. and undaresan. M. 985 The Valuaion of Opions on uures Conracs Journal of inance 40 pp Tripp J. 005 The Oregonian Porland Ore. Managing Your Money column Knigh Ridder Tribune Business News Washingon Jul 4. pp. 7. Turan G. B. and Ahme K. 000 Pricing Eurodollar uures Opions using he BDT Term rucure Model : he Effec of Yield Curve moohing Journal of uures Markes Vol. 0 pp hlfelder E. 004 Dividend huners Insiuional Invesor New York: Dec p Yadav P. K. and Pope P ock Index uures Pricing: Inernaional Evidence Journal of uures Markes 0 pp Yadav P. K. Pope P.. and Paudyal K. 994 Threshold Auoregressive Modeling in inance: he Price Differences of Equivalen Asse Mahemaical inance 4 pp Yan X. 00 Valuaion of Commodiy Derivaives in a New Muli-acor Model Review of Derivaives Research