Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics
Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se Maemaisk saisik Maemaiska insiuionen Sockholms universie 106 91 Sockholm Maemaiska insiuionen
Mahemaical Saisics Sockholm Universiy Maser hesis 2014:2 hp://www.mah.su.se Modeling VIX Fuures and Pricing VIX Opions in he Jump Diffusion Modeling Faemeh Aramian April 2014 Absrac In his hesis, a closed-form soluion for he price of opions on VIX fuures is derived by developing a erm-srucure model for VIX fuures. We analyze he VIX fuures by he Meron Jump Diffusion model and allow for sochasic ineres raes in he model. he performance of he model is invesigaed based on he daily VIX fuures prices from he Chicago Board Opion Exchange (CBOE) daa. Also, he model parameers are esimaed and opion prices are calculaed based on he esimaed values. he resuls imply ha his model is appropriae for he analysis of VIX fuures and is able o capure he empirical feaures of he VIX fuures reurns such as posiive skewness, excess kurosis and decreasing volailiy for long-erm expiraion. Posal address: Mahemaical Saisics, Sockholm Universiy, SE-106 91, Sweden. E-mail: faemeharamian@yahoo.com. Supervisor: Mia Hinnerich.
Acknowledgmen I would like o express my profound graiude o my supervisor, Mia Hinnerich. Her srong personaliy and knowledge helped me o go hrough his projec sep by sep righ from he beginning of he work. Special hanks o Bujar Huskaj and Dong Zhang for heir useful advises in his projec. I wan o hank my husband who always suppor me in my sudy and he whole life. 2
able of Conens Conens Pages 1. Inroducion 4 2. Conceps in Finance and Probabiliy heory 5 3. he VIX Fuures Model 7 3.1. VIX Fuures Model 7 3.2. Healh-Jarrow-Moron Drif Condiion 8 4. Pricing VIX Opions 9 4.1. Sochasic Ineres Rae wihou Jump 10 4.2. Sochasic Ineres Rae wih Jump 14 4.3. Consan Ineres Rae 15 5. Empirical Resuls 16 5.1. Daa 16 5.2. Empirical Properies of VIX Fuures 16 5.3. Parameer Esimaion 18 6. Conclusion 20 7. Reference 21 8. Appendix 23 3
1. Inroducion he Chicago Board Opions Exchange (CBOE) inroduced VIX fuures and VIX opions conracs for he firs ime in March 2004 and February 2006 respecively. Since 2004, he CBOE Fuures Exchange has experienced a seady progress in rading VIX fuures conracs. his growh is a consequence of acceping he volailiy as a rading insrumen and asse class by marke paricipans. Currenly, average daily volume for he VIX fuures conracs is equivalen o he fuures markes which have been around for decades. Since VIX fuures and opions are wo derivaives having VIX as he underlying and he VIX index is also ranslaed as he expeced movemen in he S&P 500 index over he nex 30-day period, he price of VIX fuures and opions are based on he expeced volailiy of he S&P 500 over he 30 day period. As a resul, Lin and Chang (2009) saed ha pricing opions on VIX fuures is more appropriae han on VIX iself. VIX fuures and opions are exchange-raded derivaives and provide he opporuniy o he invesors o rade he volailiy. Furhermore, hey are considered as a useful ool o hedge he porfolio agains fuure movemens in volailiy. he VIX opions offer he abiliy o hedge an equiy porfolio beer han oher index opions, even producs ha rade based on a porfolio s benchmark index direcly. he VIX fuures reurns have some imporan empirical feaures such as excess kurosis and posiive skewness. herefore, a proper model should be proposed o capure all hese characerisics. A large number of sudies have been currenly concenraed on VIX fuures and opions pricing. hese sudies can be divided ino wo differen caegories. In he firs caegory, differen models were developed for he VIX index in order o deermine he price of VIX fuures and opions (Psychoyios, & Skiadopoulus, (2007); Dopoye, Diagler, & Chen, (2011), Psychoyios, Dosis, & Markellos, (2009 & 2010)). Also, some sudies derived he price of VIX fuures and opions based on he model for insananeous variance of S&P 500 Index, evaluaing he VIX fuures from he S&P 500 price dynamics, (Lin, (2007, Lu, Zhu, (2010); Zhang, Shu, & Brenner, (2010); Zhang and Zhu, (2006); Zhu and Zhang, (2007); Sepp, (2008)). In he sudy by Psychoyios, Dosis, & Markellos, (2009), he VIX index is modelled by he mean-revering logarihmic diffusion model wih jump. hey evaluaed he performance based on he empirical sudy and conclude ha he behavior of VIX can be properly modelled. Laer on in 2010, hey performed a comparison beween he wo coninuous ime diffusion and jump diffusion models and sudy he behavior of he models o capure he dynamics of implied volailiy over ime. Based on heir empirical invesigaion, hey concluded ha adding jump is crucial o correcly capure he dynamics. As hey expeced, he model considering jump have a superior performance in predicing he price of he VIX fuures. In he second caegory, more effors have been carried ou o model he VIX fuures considering heir dynamics exogenously insead of focusing on he VIX iself (Huskaj and Nossman (2013), Lin (2013)). Huskaj and Nossman (2013) invesigaed he erm-srucure model for VIX fuures. heir model was a one facor model where he VIX fuures prices follow he Normal Inverse Gaussian process (NIG). hey illusraed ha his model leads o a beer fi han by jus assuming a Wiener process in he VIX fuures dynamics. In he presen sudy, we develop a closed-form soluion for he price of opions on VIX fuures by considering a erm-srucure model for VIX fuures. We model he VIX fuures by he Meron Jump 4
Diffusion model and allow for sochasic ineres raes in he model. he performance of he model is invesigaed based on he daily VIX fuures prices from he Chicago Board Opion Exchange (CBOE) for he period March 2004 o December 2010. Also, he model parameers are esimaed and opion prices are calculaed based on he esimaed values. he resuls imply ha his model is appropriae for he analysis of VIX fuures and is able o capure he empirical feaures of he VIX fuures reurns such as posiive skewness, excess kurosis and decreasing volailiy for long-erm expiraion. In fac, he main purpose of his hesis is o find an analyic formula for opion price. Moreover, o invesigae he influence of adding jump o he diffusion model o capure he empirical characerisics of VIX fuures reurns. Indeed, we modeled he VIX fuures insead of VIX iself as in previous lieraures. he res of he hesis is organized as follows: in secion 2, some of he conceps and heorems in finance and probabiliy heory are provided. In secion 3, he model and is assumpions are described. Also, he Heah-Jarrow-Moron drif condiion is derived. In secion 4, he heoreical resuls for opion pricing are provided for boh having sochasic and consan ineres rae in he model. Finally in secion 5 and 6 he empirical resuls and conclusion will be expressed respecively. 2. Conceps in Finance and Probabiliy heory In his secion, some of he definiions and heorems relaed o his hesis ha will be used in he following secions are presened in heir general forms. VIX Index: VIX is a symbol for he CBOE Marke Volailiy Index and is a measure for he volailiy of S&P 500 index opion. I represens he marke s expecaion of he movemens in he S&P 500 over he nex 30- day period. I is saed ha here is an inverse relaionship beween he movemen direcion of he SPX index and he VIX index. VIX can be calculaed heoreically by using a formula provided by he CBOE. Where, VIX 2 = 2 τ K i K i 2 i Q(K i ) 1 τ (F ( + τ) K 0 1) τ = 30 365, Q(K i ) is he price of he ou-he-money S&P 500 index opion wih srike price K i. K 0 sands for he highes exercise price less han he index forward price F ( + τ). I should be noiced ha VIX index is quoed as percenage raher han a dollar amoun. [9] 2 VIX opions: A VIX opion is an opion using he CBOE Volailiy as he underlying asse. his is he firs exchangeraded opion giving individual invesors he abiliy o rade marke volailiy. [14] 5
Fuures Conracs: A fuures conrac wih expiraion dae, on VIX as underlying is a financial derivaive wih he following properies: [1] (1) A every poin of ime 0, here exiss a quoed price F(;, VIX) in he marke, known as he fuures price a, for delivery a. (2) During any arbirary ime inerval (s, ] he holder of he conrac receives he amoun F(;, VIX) F(s;, VIX) (3) A any poin of ime prior o delivery, he spo price of he fuures conrac is equal o zero. Also, by a proposiion presened in [1], if marke prices are obained from he fixed risk neural maringale measure Q. hen, he fuures price process is given by: Noe, fuures prices are Q-maringales. F(;, VIX) = E Q [VIX ] he Likelihood Process: he following definiion can be found in [1]. Consider a filered probabiliy space (Ω, F, P, F) on a compac inerval [0, ]. Suppose now L is some nonnegaive inegrable random variable in F. Define a new measure Q on F by seing And if dq = L dp E P [L ] = 1 on F he new measure will also be a probabiliy measure. he likelihood process {L : 0 } for he measure ransformaion from P o he new probabiliy measure Q is defined as: Where L is a P-maringale and Q P. L = dq dp, on F Girsanov heorem in he jump diffusion model: he following heorem is saed in [3], Consider he filered probabiliy space (Ω, F, P, F) and assume ha N 1,. N k are opional couning process wih predicable inensiies λ 1,. λ k. Assume 6
furhermore ha W 1,.. W d are sandard independen P-Wiener processes. Le h 1,. h k be predicable process wih h i < 1, i = 1, k, P a. s, And le g 1, g d be opional processes. he likelihood process L is defined as: d k { dl = L i i=1 g dw i + L j=1 h {dn i λ i d} } (1) L 0 = 1 hen, dw i = g i d + dw Q,i, i = 1,.. d (2) λ Q,i = λ i (1 + h i ), i = 1,. k (3) Where W Q,1, W Q,d are Q -wiener processes and λ Q,i is he Q-inensiy of N i. 3. VIX Fuures Model In he presen secion, firs he VIX fuures model is presened and i is followed by deriving he Heah- Jarrow-Moron drif condiion. 3.1. VIX Fuures Model Consider a filered probabiliy space (Ω, F, P, F) ha carries a 2-mulidimensional sandard Wiener process W consising of wo independen scalar Wiener process, and a Poisson process N (wih consan inensiy λ P ). he compensaed Poisson process under P, N is defined as N = N λ P and is a P- maringale. Also, i is assumed, he model has sochasic ineres rae. he fuures conracs are wrien on VIX wih differen mauriies. he price of VIX fuures a ime wih mauriy is denoed by F(, ). Shor rae is presened by r()= f(, ), where f(, ) is forward rae. Furhermore, he bond marke is considered and we denoe he price a ime of a zero coupon bond wih expiraion dae by P(, ). he relaionship beween forward rae and -bond is defined as: f(, )= lnp(, ) he money accoun is also expressed as B()= exp ( r s ds). I is assumed ha he marke is free of arbirage and for he money accoun as numeraire, he probabiliy measure Q is a maringale measure. he dynamics of a VIX fuures conrac wih mauriy under he physical probabiliy measure P is assumed o be: 0 df(,) F(,) = α(, )d+σ(, )dw + (y 1)dN (4) 7
Which can also be wrien as: Where df(,) F(,) = (α(, ) + mλp )d+σ(, )dw + (y 1)dN (5) m = E[(y 1)] is mean of relaive jump size. In fac, (y 1) is relaive price jump size which is a lognormally disribued random variable. (y 1) i. i. d. log normal(m, e 2μ+δ2 (e δ2 1)) Also, σ(, ) is a 2-dimenional vecor known o be a deerminisic volailiy of fuures prices. α(, ) is inerpreed as deerminisic mean rae of reurn of fuures prices beween jumps and (α(, ) + mλ P ) is mean rae of reurn including jumps. Also, N, W, y are independen in he model. Moreover, he dynamics of he shor rae and he dynamics of he -bond under he assumpion of exising and non-exising jump in heir P-dynamics are assumed o be: Where dr() = α r ()d + σ r ()dw (6) dr = α r ()d + σ r ()dw + (g 1 )dn (7) P(, ) = α p (, )P(, )d + σ p (, )P(, )dw (8) P(, ) = α p (, )P(, )d + σ p (, )P(, )dw + (H 1)P(, )dn (9) (g 1) and (H 1) are relaive price jump size for he shor rae and -bonds respecively. Also, α r () in (6) and (7) is he drif erm and σ r () is a 2-dimenional deerminisic volailiy vecor of he shor rae. α p (, ) and σ p (, ) in he equaion (8) are deerminisic mean rae of reurn of -bond prices and he 2-dimenional volailiy vecor of -bond prices respecively. α p (, ) in (9) is deerminisic oal mean rae of reurn of -bond prices. 3.2. Heah-Jarrow-Moron Drif Condiion In order o derive he HJM drif condiion, ransformaion from he probabiliy measure P o Q is performed. By insering equaion (2) ino (4), compensaing he Poisson process N under probabiliy measure Q and using (3), he Q dynamics of he VIX fuures price is defined as: df(,) F(,) = [α(, ) + σ(, ) g ]d + σ(, )dw Q + (y 1)(dN (1 + h )λ P d) + (y 1)(1 + h )λ P d = [α(, ) + σ(, ) g + m(1 + h )λ P ]d + σ(, )dw Q + (y 1)dN Where dn Q is Maringale incremen under Q, g is 2-dimensional Girsanov Kernel and ( ) is a symbol for he scalar produc of he wo vecors. 8 Q
Since he fuures price is a Q-maringale, he drif erm has o be equal o zero. [α(, ) + σ(, ) g + m(1 + h )λ P ] = 0 α(, ) + mλ P = σ(, ) g mh λ P α(, ) + mλ P = σ(, ) φ + mγ Where φ denoes he 2-dimenional vecor of marke price of diffusion risk and γ denoes he marke price of jump risk. Marke price of diffusion risk and marke price of jump risk are relaed o heir Girsanov kernel g and h [3] as follows g = φ h = γ λ P Hence, he HJM drif condiion is: α(, ) + mλ P = σ(, ) φ + mγ herefore: he Q -dynamics of a VIX fuures conrac wih expiraion is: df(,) F(,) = σ(, )dw Q + (y 1)dN Q (10) Which based on he definiion of dn Q, i can also be wrien as: df(,) F(,) = λq m d + σ(, )dw Q + (y 1)dN (11) Where m = E[(y 1)] = e μ+δ2 2 1. 4. Opion Pricing In his secion, he opion price formula is derived for hree cases. Firs, pricing opions under he assumpion of having sochasic ineres rae wihou exising jump in is dynamics, second, sochasic ineres rae wih jump and he las case is pricing formula wih consan ineres rae. 9
4.1. Sochasic Ineres Rae wihou Jump Since he shor rae is sochasic in he model; he -forward measure is used o derive he opion price formula. In fac, by changing he numeraire from he money accoun in he probabiliy measure Q o he -bond in Q, he Q dynamics of VIX fuures price is obained. In order o have he Q dynamics of VIX fuures prices, he likelihood process is defined as: L = P(,) P(0,)B(), L = dq dq, on F he L -dynamic is obained by applying he Io formula o L and based on he assumpion of no having jump in he shor rae, he value of h in he equaion (1) is equal o zero. dl = σ p (, )L dw Q dw Q = σ p (, )d + dw herefore, by ransforming from Q o Q he inensiy does no change ( λ = λ Q (1 + h ) = λ Q ) and by applying he Girsanov heorem o he equaion (11): df(, ) F(, ) = σ(, ) (σp (, )d + dw ) λ Q md + (y 1)dN = ( λ Q m + σ(, ) σ p (, ))d + σ(, )dw + (y 1)dN For simpliciy in derivaion, define he scalar produc σ(, ) σ p (, ) = α F (, ) and he inensiy λ Q = λ = λ. Hence, he Q -dynamics of VIX fuures price and is price formula are: df(,) F(,) = ( λm + αf (, ) )d + σ(, )dw + (y 1)dN (12) F(, ) = F(, )exp [( λmτ + α F (s, )ds Where Y k = log(y ) i. i. d. N(μ, δ 2 ) and τ =. 1 2 σ(s, ) 2 ds ) + σ(s, )dw s he deailed derivaion of he formula (13) is presened in he Appendix. N + τ k=0 Y k ] (13) heorem: he price a ime of a European call opion wih mauriy dae and srike price K, wrien on he erminal fuures price of fuures conrac F(, ) following jump diffusion model, a any ime is given by: 10
C(, ) = P(, ) e λτ (λτ) j j 0 j! [F(, ) exp ( λmτ + α F (s, )ds + jμ + j δ2 2 ) Φ(d 1) KΦ(d 2 )] Where Φ(. ) is he cumulaive disribuion funcion of he sandard normal disribuion and d 2 = ln(f(,) d 1 = d 2 + K ) +[ λmτ+ αf (s,)ds 1 2 σ(s,) 2 ds]+jμ σ(s,) 2 ds+jδ 2 σ(s, ) 2 ds + jδ 2 Proof An arbirage-free price of a European call opion wih mauriy, wrien on he erminal fuures price of a fuures conrac F (,) and srike price K a any ime wih informaion F is given by: C(, ) = P(, )E [max(f (, ) K, 0) F )] (14) By insering he equaion (13) ino (14) and condiion on he number of jumps as: N τ = j, j = 0,1,2, he equaion (14) is expressed as: C(, ) = P(, ) E [max(f (, ) K, 0) F )] = P(, )E [(F(, ) K) I F(,)>K F ] = P(, ) j 0 Q (N τ = j) {E [F(, )exp [( λmτ + α F (s, ) ds 1 2 σ(s, ) 2 ds ) + j σ(s, )dw s + k=1 Y k ] I F(,)>K F, N τ = j ] E [KI F(,)>K F, N τ = j]} (15) Noice, σ(s, )dw s j k=0 Y k ~i. i. d. N(jμ, jδ 2 ). Hence, X= σ(s, )dw s is normally disribued wih zero mean and variance j + k=0 Y k ~N(jμ, β + jδ 2 ) where β = σ(s, ) 2 ds. Random variable X can also be presened as: σ(s, ) 2 ds and X d jμ + β + jδ 2 Z where Z is sandard normal disribued. (Z~N(0,1)) In order o calculae he equaion (15), each par of i, is compued separaely. he firs expecaion in (15) is obained as: 11
E [F(, )exp [( λmτ + α F (s, )ds j k=0 Y k 1 2 σ(s, ) 2 ds ) + σ(s, )dw s + ] I F(,)>K F, N τ = j] = E [F(, )exp [( λmτ + α F (s, )ds 1 2 σ(s, ) 2 ds ) + X] I F(,)>K F, N τ = j] = E [F(, ) exp [ λmτ + α F (s, )ds 1 2 σ(s, ) 2 ds + jμ + β + jδ 2 Z] I F(,)>K F, N τ = j] = F(, ) exp [ λmτ + α F (s, )ds 1 2 σ(s, ) 2 ds + jμ] E [exp( β + jδ 2 Z) I F(,)>K F, N τ = j] = F(, ) exp [ λmτ + α F (s, )ds 1 2 σ(s, ) 2 ds + jμ] (e β+jδ2 z f(z) dz = d 2 F(, ) exp [ λmτ + α F (s, )ds 1 2 σ(s, ) 2 ds + jμ] (e β+jδ2 z )dz. (16) d 2 1 2π e z2 2 Where f(z) = 1 2π e z2 2 is he densiy funcion of a sandard normally disribued variable Z. Also, in order o find he inegraion inerval, he indicaor funcion I F(,)>K implies ha we need o find he area ha F(, ) > K. [F(, ) N τ = j ] > K Implies: K β + jδ 2 Z > ln ( ) [ λmτ + F(,) αf (s, )ds 1 2 σ(s, ) 2 ds + jμ] Z > ln( K ) [ λmτ+ F(,) αf (s,) 1 2 σ(s,) 2 ds+jμ] herefore Z > d 2 where d 2 = ln(f(,) β+jδ 2 = d 2 K ) +[ λmτ+ αf (s,)ds 1 2 σ(s,) 2 ds]+jμ β+jδ 2 he inegral in he formula (16) is obained as: 1 (e β+jδ2 z 2π e z2 2)dz = 1 d 2 2π z z 2 e β+jδ2 2 dz = 1 d 2 2π = 1 2π e 1 2 [(z β+jδ2 ) 2 (β+jδ 2 )] dz = 1 d 2 12 2π e1 2 d 2 = e 1 2 (β+jδ2) [1 Φ ( d 2 β + jδ 2 )] = e 1 2 (β+jδ2) Φ(d 1 ) e 1 2 [z2 2 β+jδ 2 z ] dz (β+jδ2) e 1 2 [z β+jδ2 ] 2 Where Φ(. ) is he cumulaive disribuion funcion of he sandard normal random variable and d 1 = d 2 + β + jδ 2 = ln(f(,) Consequenly, he equaion (16) equals: K ) +[ λmτ+ αf (s,)ds+ 1 2 σ(s,) 2 ds]+jμ+jδ 2 β+jδ 2 d 2 dz
F(, ) exp [ λmτ + α F (s, )ds 1 2 σ(s, ) 2 ds + jμ] e 1 2 (β+jδ2) Φ(d 1 ) Also, he second expecaion in (15) is compued as: 1 2π e z2 2 KE [I F(,)>K F, N τ = j ] = K dz= KΦ(d d 2 2 ) hus, he equaion (15) is wrien as: C(, ) = P(, ) σ(s, )dw s e λτ (λτ) j j 0 j! {E [F(, )exp [( λmτ + α F (s, )ds j + k=0 Y k ] I F(,)>K F, N τ = j ] E [KI F(,)>K F, N τ = j ]} = e λτ (λτ) j 1 2 σ(s, ) 2 ds ) + P(, ) j 0 [F(, ) exp [ λmτ + αf (s, )ds 1 2 σ(s, ) 2 ds + j! jμ] e 1 2 (β+jδ2) Φ(d 1 )] P(, ) e λτ (λτ) j j 0 KΦ(d 2 ) j! herefore, he opion price formula is: C(, ) = P(, ) e λτ (λτ) j j 0 j! [F(, ) exp ( λmτ + α F (s, )ds + jμ + j δ2 2 ) Φ(d 1) KΦ(d 2 )] Where Φ(. ) is he cumulaive disribuion funcion of he sandard normal disribuion and d 2 = ln(f(,) d 1 = d 2 + K ) +[ λmτ+ αf (s,)ds 1 2 σ(s,) 2 ds]+jμ σ(s,) 2 ds+jδ 2 σ(s, ) 2 ds + jδ 2 13
4.2. Sochasic Ineres Rae wih Jump In he case ha jump exiss in he bond marke, he Q dynamics of a fuures conracs mauring a is defined as: df(,) F(,) = λq md + σ(, )dw Q + (y 1)dN (17) I is assumed ha boh bond and VIX fuures markes follow he same Poisson process wih he same inensiy. he likelihood process and is dynamic are obained as: L = P(,) P(0,)B(), L = dq dq, on F dl = σ p (, )L dw Q Q + (H 1)L dn In his case h in he formula (3) equals h = (H 1) and he inensiy of he Poisson process under Q is λ = H λ Q. By insering he equaion (2) ino (17) and compensaing for he Poisson process N under Q, he Q dynamics of VIX fuures is presened as: df(, ) F(, ) = [σ(, ) σp (, ) λ Q m]d + σ(, )dw + (y 1)dN Where N is Poisson process wih inensiy λ = H λ Q and for simpliciy [σ(, ) σ p (, )] is defined as α F (, ). Alhough he mehod of derivaion is he same as he previous case, he soluion is differen. In paricular, he Q inensiy is used when we calculae he Q probabiliy for N τ = j. C(, ) = P(, ) E [max(f (, ) K, 0) F )] = P(, )E [(F(, ) K) I F(,)>K F ] = P(, ) j 0 Q (N τ = j) {E [F(, )exp [( λ Q mτ + α F (s, )ds 1 2 σ(s, ) 2 ds ) + j σ(s, )dw s + k=0 Y k ] I F(,)>K F, N τ = j] E [K I F(,)>K F, N τ = j]} = e λτ (λ τ) j P(, ) j 0 {E [F(, )exp [( λqmτ + j! αf (s, )ds 1 2 σ(s, ) 2 ds ) + j σ(s, )dw s + k=0 Y k ] I F(,)>K F, N τ = j] E [K I F(,)>K F, N τ = j]}. Hence, by compuing he above expecaions, he price of a European call opion a ime, wih expiraion dae is defined as: C(, ) = P(, ) e λτ (λ τ) j j 0 j! [F(, ) exp ( λ Q mτ + α F (s, )ds + jμ + j δ2 2 ) Φ(d 1) KΦ(d 2 )] 14
Where Φ(. ) is he cumulaive disribuion funcion of he sandard normal disribuion and d 2 = ln(f(,) d 1 = d 2 + K ) +[ λmτ+ αf (s,)ds 1 2 σ(s,) 2 ds]+jμ σ(s,) 2 ds+jδ 2 σ(s, ) 2 ds + jδ 2 4.3. Consan Ineres rae In case of having consan ineres rae, he opion price is obained under he risk neural probabiliy measure Q and having he bank accoun as numeraire: df(,) = σ(, )dwq + (y F(,) 1)dN Q = λ Q m d + σ(, )dw Q + (y 1)dN In his case he α F (, ) = 0 in he formula (12). C(, ) = e r( ) e λqτ (λ Q τ) j j 0 j! [F(, ) exp ( λ Q mτ + jμ + j δ2 2 ) Φ(d 1) KΦ(d 2 )] Where Φ(. ) is he cumulaive disribuion funcion of a sandard normal random variable. d 2 = ln(f(,) K ) +[ λq mτ 1 2 σ(s,) 2ds ]+jμ σ(s,) 2 ds, d 1 = d 2 + σ(s, ) 2 ds + jδ +jδ 2 2 15
5. Empirical Discussion and Resuls 5.1. Daa he daily selemen prices of VIX fuures wih expiraion up o six monhs over he period March 2004 o December 2010 were gahered from he CBOE websie. We only considered conracs wih mauriy up o six monhs since longer conracs are less liquid. his resuled in a oal of 7121 observaions. he VIX Special Opening Quoe prices were muliplying by en prior o March 26, 2007 in order o deermine is final selemen value. Since ha dae, he final selemen values for VIX fuures have been based on he acual underlying index level insead of en imes he underlying index level. Hence, we divided he selemen prices from 2004 o March 26, 2007 by en o be able o work wih prices for he whole period. he opening hours of he VIX fuures markes are on business days from 7:20 A.M. o 13:15 P.M. while he majoriy of fuures markes are open almos 24 hours a day. 5.2. Empirical Properies of VIX Fuures As I menioned in he inroducion, he VIX fuures reurns have some imporan characerisics such as posiive skewness, excess kurosis and a decreasing volailiy erm srucure for long erm expiraions. hese characerisics are illusraed in able 1 where he four momens of he VIX fuures logarihmic reurns (mean, sandard deviaion, skewness and kurosis) for all sample daa and hree expiraion caegories are calculaed. Posiive and significan values of skewness and kurosis admi he exisence of hese feaures. herefore, i is saed ha he VIX fuures reurn are no normally disribued and a more appropriae and flexible erm-srucure model is needed o capure hese feaures of VIX fuures reurns. Also, i is observed ha volailiy of VIX fuures reurn decreases as here is more ime lef o mauriy. Furhermore, from he values in he able, i is clear ha mean reurns of VIX fuures are posiive for long-erm and negaive for shor-erm VIX fuures conracs. able I Descripive Saisics for he VIX Fuures Reurns ime-o All 1-2 monhs 3-4 monhs 5-6 monhs Mauriy Mean -0.0226-0.37-0.0282 0.11 Sandard deviaion 0.3663 0.29516 0.2681 0.251 Skewness 0.4322 0.0025 0.0079 0.5851 Kurosis 9.775 5.33456 5.18602 4.89958 *In his able he descripive saisics for he logarihmic reurns of VIX fuures obained from heir selemen prices is provided. he Selemen prices are from he period March 26, 2004 o December 1, 2010 and he number of daa for he whole period is 7121. he sandard deviaion is annualized by a facor 252 and average reurn is on daily based and muliplied by 100. I is observed in he able, he value of skewness and kurosis are significanly high and posiive and here is he leas volailiy for long-erm expiraion. 16
he desired candidae model for VIX fuures reurns is jump diffusion model. he Kernel densiy of VIX fuures reurns for he daa and he model wih normal disribuion are provided in Figure 1. In figure 2, he Kernel esimae of he logarihmic VIX fuures reurns ogeher wih he MJD model are observed. From he figure 2, i is clear ha he jump diffusion model provides a god fi for he sample and has a beer performance compared o a case wihou jump. FIGURE 1 Kernel Esimae of VIX Fuures reurns and Normal FIGURE 2 Kernel Esimae of Logarihmic VIX Fuures reurns and MJD 17
5.3. Parameers Esimaion here exis differen mehods for he purpose of parameer esimaion. Alhough Maximum Likelihood Esimaion is one of he mos popular mehods, in he case of jump diffusion model, i does no work well and i is no a careful numerical opimizaion. he reason is ha he maximum likelihood is very sensiive o he iniial values and by really small changes in hose values, he likelihood funcion canno be converges easily 2. herefore, o esimae he parameers of he model, he Non-Linear Leas Square (NLS) mehod is used and hey are esimaed under he assumpion ha he model has consan ineres rae and one dimensional Wiener process. Also, i is assumed ha he Girsanov Kernel h (in he equaion (3)) is equal o zero. herefore, based on he relaionship beween he marke price of jump risk and is Kernel, he marke price of jump risk is zero in our esimaion. During he process of esimaion by NLS, i was observed ha by changing he iniial values, he model converges o differen esimaed values. Consequenly, i was clear, here are more han one local minimum ha minimize he error beween he daa and he model. In fac, he global minimum should be considered o esimae he parameers. he Meron Jump Diffusion model is he mixure of N normally disribued erms and he mean, variance and weigh of j h sochasic variable in he mixure are m j = (α σ2 ) τ + jμ, s 2 j 2 = σ 2 τ + jδ 2 and w j = e λτ (λτ) j respecively. he sufficienly large N is j! chosen and i should be noed ha he seleced N depends on λ. he numerical sudies using daily observaions demonsrae ha here is no significan difference in esimaes from N=20. For his sudy he number of jumps is considered o be N=140.he volailiy in he model in he equaion (4) is specified as σ(, ) = σ 1 e σ 2( ) where σ 1 and σ 2 are nonnegaive 3. Also, he marke price of risk is assumed o be consan no ime dependen. In able 2, he esimaed values of he six parameers σ 1, σ 2,μ, δ, λ and φ of he model are observed where μ and δ are he mean and sandard deviaion of logarihmic jump size, λ is he P-inensiy and φ is marke price of diffusion risk. able ΙΙ Esimaion Resuls for he VIX Fuures Models Models σ 1 σ 2 μ δ λ φ Meron Jump 0.1836 0.033 0.0003 0.0285 252.0681-3.3139 Diffusion Normal case 0.3561 0.1257 - - - -2.1312 *he models parameers are esimaed using daily logarihmic reurns of VIX fuures prices wih mauriy up o six monhs over he period March 26, 2004 o December 2010. he number of daa is 7121. 2 Some empirical researches have applied mehod oher han Maximum Likelihood Esimaion. Duncan and Randal (2009) is one of he sudies used EM algorihm for esimaion. 3 his volailiy funcion was suggesed by Hilliard-Reise (1998) 18
Since he inensiy is he expeced number of jumps, is larger value resuls in occurring jump more frequenly. Moreover, he sign of μ (he mean of logarihmic jump size) deermines if reurns are posiive or negaive skewed. From he able, i is observed ha μ is posiive for our daa which admis he posiive skewness feaure of he VIX fuures reurns. able illusraes ha marke price of risk has a negaive sign for boh MJD model and Normal model which is consisen wih he resuls in he sudy by Nossman & Wilhelmsson (2008). Figure 3 illusraes he changes in he call opion values in boh he MJD model and he sandard model wihou jump for differen ime o mauriies. he following assumpions are considered, namely ineres rae, r =0.075 and curren VIX fuures price, F=30 1 Week o Mauriy 1 Monh o Mauriy 3 Monhs o Mauriy 5 Monhs o Mauriy Figure 3 19 MJD Call Price vs. Normal Call Price
he changes in he price of call opions wih respec o he srike price are illusraed for boh models in figure 3. he figures demonsrae ha he MJD call prices have greaer values han he sandard model for boh in-he-money and ou-he-money opions. Also, i is observed, by increasing mauriy hese resuls sill hold. his conclusion is consisen wih he resuls in he research by Masuda (2004) who compared he price of sock call opions in he MJD model and he Black-Scholes model. Moreover, figures illusrae ha by increasing expiraion ime he price difference beween he MJD call price and he Black call price increases. 6. Conclusion I is around a decade ha VIX fuures and opions have been presened o he marke and are rading in a large volume oday. he lieraures on VIX fuures and opions are growing speedily. A large number of researches have been done o reveal differen characerisics of VIX fuures. Some of he researches focus on modeling he VIX index and ry o find an appropriae disribuion for VIX fuures reurns while oher researchers specified he VIX fuures dynamics exogenously in heir sudies. In his hesis, in he heory par, he VIX fuures were modeled by he Meron jump diffusion model and a closed-form soluion for he price of opions on VIX fuures was derived for boh sochasic and consan ineres rae cases in he model. In he empirical par, by using he hisorical VIX fuures prices from he CBOE daa, he behaviors of he VIX fuures reurns were invesigaed and he model parameers were esimaed. he descripive saisics of he daa illusraed ha he VIX fuures reurns are posiive skewed and have excess kurosis. herefore, i is clear ha he VIX fuures reurns are no normally disribued. Also, we calculaed he price of he VIX call opions for boh he MJD model and he sandards model using he esimaed parameers. he resuls implied ha he MJD lead o greaer values han he oher model for boh in-hemoney and ou-he-money opions. Hence, i is concluded ha adding jump o he diffusion process is crucial o capure he feaures of he daa. In fac, he jump diffusion model is well approximaed and presens beer performance compared o he sandard case. In order o exend his sudy, he performance of he model can be assessed by using he marke VIX opions. Also, by applying differen models o he VIX fuures and invesigaing he performance of ha model in fuure researches, he mos appropriae and fi model can be revealed. 20
7. Reference [1] Björk,. (2009). Arbirage heory in Coninuous ime. (3 rd Ed.). New York: Oxford Universiy Press. [2] Björk,., & Landen, C. (2000). On he erm Srucure of Fuures and Forward Prices. SSE/EFI Working Paper Series in Economics and Finance, No. 417. [3] Björk,. (2011). An Inroducion o Poin Process from a Maringale Poin of View. Lecure noe. KH. [4] Black, F., & Scholes, M. (1973). he pricing of opions and corporae liabiliies. Journal of Poliical Economy, 81, 637 659 [5] Dosis, G., Psychoyios, D., & Skiadopoulus, G. (2007). An empirical comparison of coninuous-ime models of implied volailiy indices. Journal of Banking and Finance, 31, 3584 3603. [6] Haskay, B. and M. Nossman, (2013), A erm Srucure Model for VIX Fuures. Journal of Fuures Markes, Vol. 33, Iss. 5, PP. 421-442 [7] Hilliard, J., and Reis, J. (1998). Valuaion of commodiy fuures and opions under sochasic convenience yields, ineres raes, and jump diffusion in he spo. JFQA 33, 1, 61-86. [8] Honore, P. (1998). Pifalls in Esimaing Jump Diffusion Model. Social Science Research Nework. [9] Lin, Y. N. (2007). Pricing VIX fuures: Evidence form inegraed physical and risk neural probabiliy measures. he Journal of Fuures Markes, 27, 1175 1217. [10] Lin, Y. N., & Chang, C. (2009). VIX opion pricing. he Journal of Fuures Markes, 29, 523 543. [11] Lu, Z. J., & Zhu, Y. Z. (2010). Volailiy componens: he erm srucure dynamics of VIX fuures. he Journal of Fuures Markes, 30, 230 256. [12] Meron, R.C., (1975). Opion Pricing When Underlying Sock reurns are Disconinuous. Journal of Financial Economics, 3, 125-144. Q Norh-Holland Publishing Company [13] Psychoyios, D., Dosis, G., & Markellos, R. N. (2009). A jump diffusion model for VIX volailiy opions and fuures. Review of Quaniaive Finance and Accouning, 35, 245 269. [14] Rhoads, R. (2011). rading VIX Derivaives: rading and Hedging Sraegies Using VIX Fuures, Opions and Exchange raded Noes. New Jersey: John Wiley & Sons 21
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Appendix A.1. Fuures Price Formula I was saed in secion hree ha he model has he following P-dynamics: df(, ) F(, ) = α(, )d + σ(, )dw + (y 1)dN Define funcion g(, ) = ln (F(, )) and by applying he Io formula o his funcion: dg(, ) = 1 1 df(, ) (F(, F(,) F 2 (,) ))2 + dn [ln(f(, ) + F(, )(y 1)) ln(f(, ))] = (α(, ) 1 2 σ(s, ) 2 ) d + σ(, )dw + ln(y ) dn = (α(, ) 1 2 σ(s, ) 2 ) d + σ(, )dw + Y dn Where Y k = ln (y k ) By inegraing over he inerval [, ]: g(, ) = ln(f(, )) = g(, ) + (α(s, ) 1 2 σ(s, ) 2 ) ds + σ(s, )dw s Hence, F(, ) = F(, ) exp [ α(s, )ds 1 2 σ(s, ) 2 ds + σ(s, )dw s F(, ) exp [ α(s, )ds 1 2 σ(s, ) 2 ds + σ(s, )dw s N k=0 y k ] N + k=0 Y N + k=0 Y k ] = herefore, he price of fuures conrac wih expiraion a ime is calculaed by he following formula: F(, ) = F(, ) exp [ α(s, )ds 1 2 σ(s, ) 2 ds + σ(s, )dw s N + Y k ] k=0 23