Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
|
|
- Kenneth Ramsey
- 8 years ago
- Views:
Transcription
1 Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics
2 Maseruppsas 2014:2 Maemaisk saisik April Maemaisk saisik Maemaiska insiuionen Sockholms universie Sockholm Maemaiska insiuionen
3 Mahemaical Saisics Sockholm Universiy Maser hesis 2014:2 hp:// 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 , Sweden. faemeharamian@yahoo.com. Supervisor: Mia Hinnerich.
4 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
5 able of Conens Conens Pages 1. Inroducion 4 2. Conceps in Finance and Probabiliy heory 5 3. he VIX Fuures Model VIX Fuures Model Healh-Jarrow-Moron Drif Condiion 8 4. Pricing VIX Opions Sochasic Ineres Rae wihou Jump Sochasic Ineres Rae wih Jump Consan Ineres Rae Empirical Resuls Daa Empirical Properies of VIX Fuures Parameer Esimaion Conclusion Reference Appendix 23 3
6 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
7 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 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 = , 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
8 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
9 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 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
10 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 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
11 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+δ 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
12 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
13 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
14 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 π 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
15 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
16 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
17 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 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
18 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 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 Sandard deviaion Skewness Kurosis *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 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
19 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
20 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 Diffusion Normal case *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 he number of daa is 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
21 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
22 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
23 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 [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, [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, [6] Haskay, B. and M. Nossman, (2013), A erm Srucure Model for VIX Fuures. Journal of Fuures Markes, Vol. 33, Iss. 5, PP [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, [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, [10] Lin, Y. N., & Chang, C. (2009). VIX opion pricing. he Journal of Fuures Markes, 29, [11] Lu, Z. J., & Zhu, Y. Z. (2010). Volailiy componens: he erm srucure dynamics of VIX fuures. he Journal of Fuures Markes, 30, [12] Meron, R.C., (1975). Opion Pricing When Underlying Sock reurns are Disconinuous. Journal of Financial Economics, 3, 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, [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
24 [15] Wang, Z., & Daigler, R.. (2011). he performance of VIX opion pricing models: Empirical evidence beyond simulaion. he Journal of Fuures Markes, 31, [16] Zhang, J. E., Shu, J., & Brenner, M. (2010). he new marke for volailiy rading. he Journal of Fuures Markes, 30, [17] Zhang, J. E., & Zhu, Y. Z. (2006). VIX fuures. he Journal of Fuures Markes, 26,
25 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
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:
More informationPricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationA Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets
A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationOption Put-Call Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationUNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.
UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 5-7 85 Barcelona Absrac We see ha he price of an european call opion
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationSkewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance
Finance Leers, 003, (5), 6- Skewness and Kurosis Adjused Black-Scholes Model: A Noe on Hedging Performance Sami Vähämaa * Universiy of Vaasa, Finland Absrac his aricle invesigaes he dela hedging performance
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationSPEC model selection algorithm for ARCH models: an options pricing evaluation framework
Applied Financial Economics Leers, 2008, 4, 419 423 SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics,
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationPricing Futures and Futures Options with Basis Risk
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
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationHow To Price An Opion
HE PERFORMANE OF OPION PRIING MODEL ON HEDGING EXOI OPION Firs Draf: May 5 003 his Version Oc. 30 003 ommens are welcome Absrac his paper examines he empirical performance of various opion pricing models
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 8-89 ublished Online June 0 in MECS (hp://www.mecs-press.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecs-press.ne/ijem Opion ricing Under Sochasic Ineres
More informationThe Generalized Extreme Value (GEV) Distribution, Implied Tail Index and Option Pricing
he Generalized Exreme Value (GEV) Disribuion, Implied ail Index and Opion Pricing Sheri Markose and Amadeo Alenorn his version: 6 December 200 Forhcoming Spring 20 in he Journal of Derivaives Absrac Crisis
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 94-9(5)634-4 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE
More informationChapter 7. Response of First-Order RL and RC Circuits
Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationMeasuring macroeconomic volatility Applications to export revenue data, 1970-2005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationValuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate
Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and Jean-Luc Prigen hema, Universiy of Cergy-Ponoise, Cergy-Ponoise, France E-mails: fabricebarhelemy@u-cergyfr; jean-lucprigen@u-cergyfr
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationStochastic Calculus and Option Pricing
Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes
More informationDescription of the CBOE S&P 500 BuyWrite Index (BXM SM )
Descripion of he CBOE S&P 500 BuyWrie Index (BXM SM ) Inroducion. The CBOE S&P 500 BuyWrie Index (BXM) is a benchmark index designed o rack he performance of a hypoheical buy-wrie sraegy on he S&P 500
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures
More informationCredit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work
More informationDoes Option Trading Have a Pervasive Impact on Underlying Stock Prices? *
Does Opion Trading Have a Pervasive Impac on Underlying Sock Prices? * Neil D. Pearson Universiy of Illinois a Urbana-Champaign Allen M. Poeshman Universiy of Illinois a Urbana-Champaign Joshua Whie Universiy
More informationModeling VXX. First Version: June 2014 This Version: 13 September 2014
Modeling VXX Sebasian A. Gehricke Deparmen of Accounancy and Finance Oago Business School, Universiy of Oago Dunedin 9054, New Zealand Email: sebasian.gehricke@posgrad.oago.ac.nz Jin E. Zhang Deparmen
More informationDefault Risk in Equity Returns
Defaul Risk in Equiy Reurns MRI VSSLOU and YUHNG XING * BSTRCT This is he firs sudy ha uses Meron s (1974) opion pricing model o compue defaul measures for individual firms and assess he effec of defaul
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 2008-05-30 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationDOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR
Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios
More informationIndividual Health Insurance April 30, 2008 Pages 167-170
Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK-2100 Copenhagen Ø, Denmark PFA Pension,
More informationDynamic Option Adjusted Spread and the Value of Mortgage Backed Securities
Dynamic Opion Adjused Spread and he Value of Morgage Backed Securiies Mario Cerrao, Abdelmadjid Djennad Universiy of Glasgow Deparmen of Economics 27 January 2008 Absrac We exend a reduced form model for
More informationFX OPTION PRICING: RESULTS FROM BLACK SCHOLES, LOCAL VOL, QUASI Q-PHI AND STOCHASTIC Q-PHI MODELS
FX OPTION PRICING: REULT FROM BLACK CHOLE, LOCAL VOL, QUAI Q-PHI AND TOCHATIC Q-PHI MODEL Absrac Krishnamurhy Vaidyanahan 1 The paper suggess a new class of models (Q-Phi) o capure he informaion ha he
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationPricing Black-Scholes Options with Correlated Interest. Rate Risk and Credit Risk: An Extension
Pricing Black-choles Opions wih Correlaed Ineres Rae Risk and Credi Risk: An Exension zu-lang Liao a, and Hsing-Hua Huang b a irecor and Professor eparmen of inance Naional Universiy of Kaohsiung and Professor
More informationInvestor sentiment of lottery stock evidence from the Taiwan stock market
Invesmen Managemen and Financial Innovaions Volume 9 Issue 1 Yu-Min Wang (Taiwan) Chun-An Li (Taiwan) Chia-Fei Lin (Taiwan) Invesor senimen of loery sock evidence from he Taiwan sock marke Absrac This
More informationThe performance of popular stochastic volatility option pricing models during the Subprime crisis
The performance of popular sochasic volailiy opion pricing models during he Subprime crisis Thibau Moyaer 1 Mikael Peijean 2 Absrac We assess he performance of he Heson (1993), Baes (1996), and Heson and
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationJump-Diffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach
ump-diffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,
More informationEfficient Pricing of Energy Derivatives
Efficien Pricing of Energy Derivaives Anders B. Trolle EPFL and Swiss Finance Insiue March 1, 2014 Absrac I presen a racable framework, firs developed in Trolle and Schwarz (2009), for pricing energy derivaives
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationPRICING and STATIC REPLICATION of FX QUANTO OPTIONS
PRICING and STATIC REPLICATION of F QUANTO OPTIONS Fabio Mercurio Financial Models, Banca IMI 1 Inroducion 1.1 Noaion : he evaluaion ime. τ: he running ime. S τ : he price a ime τ in domesic currency of
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4-11-2008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationStochastic Volatility Models: Considerations for the Lay Actuary 1. Abstract
Sochasic Volailiy Models: Consideraions for he Lay Acuary 1 Phil Jouber Coomaren Vencaasawmy (Presened o he Finance & Invesmen Conference, 19-1 June 005) Absrac Sochasic models for asse prices processes
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationAPPLICATION OF THE KALMAN FILTER FOR ESTIMATING CONTINUOUS TIME TERM STRUCTURE MODELS: THE CASE OF UK AND GERMANY. January, 2005
APPLICATION OF THE KALMAN FILTER FOR ESTIMATING CONTINUOUS TIME TERM STRUCTURE MODELS: THE CASE OF UK AND GERMANY Somnah Chaeree* Deparmen of Economics Universiy of Glasgow January, 2005 Absrac The purpose
More informationINVESTMENT GUARANTEES IN UNIT-LINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE
INVESMEN UARANEES IN UNI-LINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN
More informationOn the Role of the Growth Optimal Portfolio in Finance
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN 1441-8010 www.qfrc.us.edu.au
More informationLECTURE 7 Interest Rate Models I: Short Rate Models
LECTURE 7 Ineres Rae Models I: Shor Rae Models Spring Term 212 MSc Financial Engineering School of Economics, Mahemaics and Saisics Birkbeck College Lecurer: Adriana Breccia email: abreccia@emsbbkacuk
More informationOptions and Volatility
Opions and Volailiy Peer A. Abken and Saika Nandi Abken and Nandi are senior economiss in he financial secion of he Alana Fed s research deparmen. V olailiy is a measure of he dispersion of an asse price
More informationCommon Risk Factors in the US Treasury and Corporate Bond Markets: An Arbitrage-free Dynamic Nelson-Siegel Modeling Approach
Common Risk Facors in he US Treasury and Corporae Bond Markes: An Arbirage-free Dynamic Nelson-Siegel Modeling Approach Jens H E Chrisensen and Jose A Lopez Federal Reserve Bank of San Francisco 101 Marke
More informationRandom Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary
Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationConceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100...
Normal (Gaussian) Disribuion Probabiliy De ensiy 0.5 0. 0.5 0. 0.05 0. 0.9 0.8 0.7 0.6? 0.5 0.4 0.3 0. 0. 0 3.6 5. 6.8 8.4 0.6 3. 4.8 6.4 8 The Black-Scholes Shl Ml Moel... pricing opions an calculaing
More informationA Tale of Two Indices
PEER CARR is he direcor of he Quaniaive Finance Research group a Bloomberg LP and he direcor of he Masers in Mahemaical Finance program a he Couran Insiue of New York Universiy NY. pcarr4@bloomberg.com
More informationOrder Flows, Delta Hedging and Exchange Rate Dynamics
rder Flows Dela Hedging and Exchange Rae Dynamics Bronka Rzepkowski # Cenre d Eudes rospecives e d Informaions Inernaionales (CEII) ABSTRACT This paper proposes a microsrucure model of he FX opions and
More informationHow Useful are the Various Volatility Estimators for Improving GARCH-based Volatility Forecasts? Evidence from the Nasdaq-100 Stock Index
Inernaional Journal of Economics and Financial Issues Vol. 4, No. 3, 04, pp.65-656 ISSN: 46-438 www.econjournals.com How Useful are he Various Volailiy Esimaors for Improving GARCH-based Volailiy Forecass?
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationSURVEYING THE RELATIONSHIP BETWEEN STOCK MARKET MAKER AND LIQUIDITY IN TEHRAN STOCK EXCHANGE COMPANIES
Inernaional Journal of Accouning Research Vol., No. 7, 4 SURVEYING THE RELATIONSHIP BETWEEN STOCK MARKET MAKER AND LIQUIDITY IN TEHRAN STOCK EXCHANGE COMPANIES Mohammad Ebrahimi Erdi, Dr. Azim Aslani,
More informationDoes Option Trading Have a Pervasive Impact on Underlying Stock Prices? *
Does Opion Trading Have a Pervasive Impac on Underlying Soc Prices? * Neil D. Pearson Universiy of Illinois a Urbana-Champaign Allen M. Poeshman Universiy of Illinois a Urbana-Champaign Joshua Whie Universiy
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of Erlangen-Nuremberg Lange Gasse
More informationT ϕ t ds t + ψ t db t,
16 PRICING II: MARTINGALE PRICING 2. Lecure II: Pricing European Derivaives 2.1. The fundamenal pricing formula for European derivaives. We coninue working wihin he Black and Scholes model inroduced in
More informationAn accurate analytical approximation for the price of a European-style arithmetic Asian option
An accurae analyical approximaion for he price of a European-syle arihmeic Asian opion David Vyncke 1, Marc Goovaers 2, Jan Dhaene 2 Absrac For discree arihmeic Asian opions he payoff depends on he price
More informationMarket Liquidity and the Impacts of the Computerized Trading System: Evidence from the Stock Exchange of Thailand
36 Invesmen Managemen and Financial Innovaions, 4/4 Marke Liquidiy and he Impacs of he Compuerized Trading Sysem: Evidence from he Sock Exchange of Thailand Sorasar Sukcharoensin 1, Pariyada Srisopisawa,
More informationForeign Exchange and Quantos
IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in
More informationSupplementary Appendix for Depression Babies: Do Macroeconomic Experiences Affect Risk-Taking?
Supplemenary Appendix for Depression Babies: Do Macroeconomic Experiences Affec Risk-Taking? Ulrike Malmendier UC Berkeley and NBER Sefan Nagel Sanford Universiy and NBER Sepember 2009 A. Deails on SCF
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process,
Chaper 19 The Black-Scholes-Vasicek Model The Black-Scholes-Vasicek model is given by a sandard ime-dependen Black-Scholes model for he sock price process S, wih ime-dependen bu deerminisic volailiy σ
More informationVolatility Forecasting Techniques and Volatility Trading: the case of currency options
Volailiy Forecasing Techniques and Volailiy Trading: he case of currency opions by Lampros Kalivas PhD Candidae, Universiy of Macedonia, MSc in Inernaional Banking and Financial Sudies, Universiy of Souhampon,
More information11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees.
The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling 1 Insiu für Finanz- und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationThe Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies
1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz- und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany
More informationAn Interest Rate Swap Volatility Index and Contract
Anonio Mele QUASaR Yoshiki Obayashi Applied Academics LLC Firs draf: November 10, 2009. This version: June 26, 2012. ABSTRACT Ineres rae volailiy and equiy volailiy evolve heerogeneously over ime, comoving
More information12. Market LIBOR Models
12. Marke LIBOR Models As was menioned already, he acronym LIBOR sands for he London Inerbank Offered Rae. I is he rae of ineres offered by banks on deposis from oher banks in eurocurrency markes. Also,
More informationThe Information Content of Implied Skewness and Kurtosis Changes Prior to Earnings Announcements for Stock and Option Returns
The Informaion Conen of Implied kewness and urosis Changes Prior o Earnings Announcemens for ock and Opion Reurns Dean Diavaopoulos Deparmen of Finance Villanova Universiy James. Doran Bank of America
More informationIMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **
IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include so-called implici or embedded opions.
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees
1 The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling Insiu für Finanz- und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationReturn Calculation of U.S. Treasury Constant Maturity Indices
Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion
More informationIntroduction to Arbitrage Pricing
Inroducion o Arbirage Pricing Marek Musiela 1 School of Mahemaics, Universiy of New Souh Wales, 252 Sydney, Ausralia Marek Rukowski 2 Insiue of Mahemaics, Poliechnika Warszawska, -661 Warszawa, Poland
More informationOptimal Longevity Hedging Strategy for Insurance. Companies Considering Basis Risk. Draft Submission to Longevity 10 Conference
Opimal Longeviy Hedging Sraegy for Insurance Companies Considering Basis Risk Draf Submission o Longeviy 10 Conference Sharon S. Yang Professor, Deparmen of Finance, Naional Cenral Universiy, Taiwan. E-mail:
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationTHE DETERMINATION OF PORT FACILITIES MANAGEMENT FEE WITH GUARANTEED VOLUME USING OPTIONS PRICING MODEL
54 Journal of Marine Science and echnology, Vol. 13, No. 1, pp. 54-60 (2005) HE DEERMINAION OF POR FACILIIES MANAGEMEN FEE WIH GUARANEED VOLUME USING OPIONS PRICING MODEL Kee-Kuo Chen Key words: build-and-lease
More informationThe naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1
Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces ime-series smoohing forecasing mehods. Various models are discussed,
More information