European Option Pricing for a Stochastic Volatility Lévy Model with Stochastic Interest Rates

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Jonal of Mahemaical Finance 98-8 doi:.436/mf..33 Pblished Online Noembe (hp://www.scirp.

1 Jonal of Mahemaical Finance 98-8 doi:.436/mf..33 Pblished Online Noembe (hp:// Eopean Opion Picing fo a Sochasic Volailiy Léy Model wih Sochasic Inees Raes Saisa Pinkham Paioe Saayaham School of Mahemaics Insie of Science Sanaee Uniesiy of echnology Nakhon Rachasima hailand saisa@mah.s.ac.h paioe@s.ac.h Receied Ags 7 ; eised Sepembe 9 ; acceped Sepembe 8 Absac We pesen a Eopean opion picing when he ndelying asse pice dynamics is goened by a linea combinaion of he ime-change Léy pocess and a sochasic inees ae which follows he Vasicek pocess. We obain an explici fomla fo he Eopean call opion in em of he chaaceisic fncion of he ail pobabiliies. Keywods: ime-change Léy Pocess Sochasic Inees Rae Vasicek Pocess Fowad Mease Opion Picing. Inodcion Le F P be a pobabiliy space. A sochasic pocess L is a Léy pocess if i has independen and saionay incemens and has a sochasically coninos sample pah i.e. fo any lim P L hl. he simh ples possible Léy pocesses ae he sandad Bownian moion Poisson pocess and compond Poisson W N i pocess Y whee is Poisson pocess wih inen- i N siy and Yi ae i.i.d. andom aiables. Of cose we can bild a new Léy pocess fom known ones by sing he echniqe of linea ansfomaion. Fo example he mp diffsion pocess N W Y whee i i ae consans is a Léy pocess which comes fom a linea ansfomaion of wo independen Léy pocesses i.e. a Bownian moion wih dif and a compond Poisson pocess. Assme ha a isk-neal pobabiliy mease exiss and all pocesses in Secion will be consideed nde his isk-neal mease. In he Black-Scholes model he pice of a isky asse S nde a isk-neal mease and wih non diidend paymen follows whee N S SexpL Sexp W (.) is a isk-fee inees aes is a o- lailiy coefficien of he sock pice. Insead of modeling he log ens L W wih a nomal disibion. We now eplace i wih a moe sophisicaed pocess L which is a Léy pocess of he fom L W J (.) whee J and denoes a pe Léy mp componen (i.e. a Léy pocess wih no Bownian moion pa) and is conexiy adsmen. We assme ha he pocesses W and J ae independen. o incopoae he olaileiy effec o he model (.) we follow he echniqe of Ca and W [] by sbodinaing a pa of a sandad Bownian moion W and a pa of mp Léy pocess J by he ime inegal of a mean eeing Cox Ingesoll Ross (CIR) pocess d s s whee follows he CIR pocess d d d W (.3) Hee W is a sandad Bownian moion which coesponds o he pocess. he consan is he ae a which he pocess ees owad is long em mean and is he olailiy coefficien of he pocess. Copyigh SciRes.

2 S. PINKHAM E AL. 99 Hence he model (.) has been changed o L W J (.4) and his new pocess is called a sochasic olailiy Ley pocess. One can inepe as he sochasic clock pocess wih aciiy ae pocess. By eplacing L in (.) wih L we obain a model of an ndelying asse nde he isk-neal mease wih sochasic olailiy as follows: S S expw J (.5) In his pape we shall conside he poblem of finding a fomla fo Eopean call opions based on he ndelying asse model (.5) fo which he consan inees aes is eplaced by he sochasic inees aes and J is compond Poisson pocess i.e. he model nde o consideaion is gien by S S exp W J (.6) Hee we assme ha follows he Vasicek pocess d d dw (.7) W is a sandad Bownian moion wih espec o he pocess and d W d d d W W W. he consan is he ae a which he inees ae ees owad is long em mean is he olailiy coefficien of he inees ae pocess (.7) he consan is a speed eesion.. Lieae Reiews Many financial engineeing sdies hae been ndeaken o modify and impoe he Black-Scholes model. Fo example he mp diffsion models of Meon [] he sochasic Volailiy mp diffsion model of Baes [3] and Yan and Hanson [4]. Fhemoe he ime change Léy models poposed by Ca and W []. he poblem of opion picing nde sochasic inees aes has been inesigaed fo along ime. Kim [5] consced he opion picing fomla based on Black-Scholes model nde seeal sochasic inees ae pocesses i.e. Vasicek CIR Ho-Lee ype. He fond ha by incpoaing sochasic inees aes ino he Black-Scholes model fo a sho maiy opion does no conibe o impoemen in he pefomance of he oiginal Black- Scholes picing fomla. Bigo and Mecio [6] menion ha he sochasic feae of inees aes has a songe impac on he opion pice when picing fo a long maiy opion. Ca and W [] conine his sdy by giing he opion picing fomla based on a ime-changed Léy pocess model. B hey sill se consan inees aes in he model. In his pape we gie an analysis on he opion picing model based on a ime-changed Léy pocess wih sochasic inees aes. he es of he pape is oganized as follows. he dynamics nde he fowad mease is descibed in Secion 3. he opion picing fomla is gien in Secion 4. Finally he close fom solion fo a Eopean call opion in ems of he chaaceisic fncion is gien in Secion he Ddynamics nde he Fowad Mease We begin by giing a bief eiew of he definiion of a coelaed Bownian moion and some of is popeies (fo moe deails one see Bmmelhis [7]). Recalling n ha a sandad Bownian moion in R is a sochasic pocess Z whose ale a ime is simply a eco of n independen Bownian moions a Z Z Z n We se Z insead of W since we wold like o esee he lae fo he moe geneal case of coelaed Bownian moion which will be defined as follows: be a (consan) posiie symmeic Le i i n maix saisfying ii and i By Cholesky s decomposiion heoem one can find an ppe iangl n n maix h i sch ha whee Η is he anspose of he maix Η. Le Z Z Zn be a sandad Bownian moion as inodced aboe we define a new eco-aled pocess W W Wn by W Z o in em of componens n Wi hiz i n he pocess W is called a coelaed Bownian moion wih a (consan) coelaion maix. Each componen pocess Wi is iself a sandad Bownian moion. Noe ha if Id (he ideniy maix) hen W is a sandad Bownian moion. Fo example if we le a symmeic maix (3.) hen has a Cholesky decomposiion of he fom HH whee H is an ppe iangla maix of he fom Copyigh SciRes.

3 S. PINKHAM E AL. H Le Z Z Z Z be hee independen Bownian moions hen W W W W defined by W Z o in ems of componens W Z Z W Z W Z (3.) Now le s n o o poblem. Noe ha by Io s lemma he model (.6) has he dynamic gien by Y ds S d dw S e d N m d d d W d d d W (3.3) whee e Y E dwdw dwdw and m dwdw d. We can e-wie he dynamic (3.3) in ems of hee independen Bownian moions Z Z Z follows (3.) we ge ds S md dz dz (3.4) Y S e d N d d d (3.5) d d d Z (3.6) his decomposiion makes i easie o pefom a mease ansfomaion. In fac fo any fixed maiy le s denoe by he -fowad mease i.e. he pobabiliy mease ha is defined by he Radon- Nikodym deiaie exp d d (3.7) d P Hee P is he pice a ime of a zeo-copon bond wih maiy and is defined as e ds s P E F (3.8) Nex Conside a coninos-ime economy whee inees aes ae sochasic and saisfy (3.5). Since he SDE (3.5) saisfies all he necessay condiions of heoem 3 see Poe [8] hen he solion of (3.5) has he Mako popey. As a conseqence he zeo copon bond pice a ime nde he mease in (3.8) saisfies Z Noe ha P E exp sds (3.9) P depends on only insead of depending on all infomaion aailable in F p o ime. As sch i becomes a fncion F of P F meaning ha he picing poblem can now be fomlaed as a seach fo he fncion F. Lemma he pice of a zeo copon bond can be deied by comping he expecaion (3.9). We obain P exp a b (3.) whee b e 3 a e e 3 Poof. See Pial [9] (pp ). Lemma he pocess following he dynamics in (3.5) can be wien in he fom whee he pocess x w fo each (3.) x saisfies dx xd d Z x. (3.) Moeoe he fncion w() is deeminisic and well defined in he ime ineal [] which saisfied w In paicla w e e. Poof. o sole he solion of SDE (3.5) g e and sing Io s Lemma Le hen g g g dg d d d de e de d Z = e d e dz d (3.3) (3.4) Inegaed on boh side he aboe eqaion fom o whee and simplified one ge e e e dz By sing he definiion of w fom (3.3) Copyigh SciRes.

4 S. PINKHAM E AL. e d w Z (3.5). Noe ha he solion of (3.) is whee e e w e e d e dz x x Z Hence w x. (3.6) fo each. he poof is now complee. Nex we shall calclae he Radon-Nikodym deiaie as appea in (3.7). By Lemma and we hae x w and P. Sbsiing and P ino (3.7) we hae x w d d exp d exp a b exp x d d e Sochasic inegaion by pas implies ha x d d d x x x (3.8) By sbsiing he expession fo dx fom (3.) dx xd dz (3.7) (3.9) Moeoe by sbsiing he expession fo x fom (3.6) he fis inegal on he igh hand side of (3.9) becomes x d (3.) s e dz d Using inegal by pas we hae (Eqaion 3.) Sbsiing (3.) ino (3.9) we obain dx e Z d o Hence d e x dz (3.) Sbsiing (3.) ino (3.7) once ge d d Z exp e d e d (3.3) he Gisano heoem hen implies ha he hee poc esses Z Z and Z defined by dz dz e d (3.4) dz d Z dz dz ae hee independen Bownian moions nde he meas- e. heefoe he dynamics of and S nde ae gien by Y e d N ds S d dz dz m S d e d d Z d d d Z. (3.5) 4. he Picing of a Eopean Call Opion on he Gien Asse Le S be he pice of a financial asse modeled as a sochasic pocess on a fileed pobabiliy space FF F is sally aken o be he pice his- oy p o ime. All pocesses in his secion will be defined in his space. We denoe C he pice a ime of a Eopean call opion on he cen pice of an ndelying asse S wih sike pice K and expiaion ime. s e dz d o e dz e d e d e dz e e d dz e dz s s e dz e d e dz d e d s s d Z. (3.) Copyigh SciRes.

5 S. PINKHAM E AL. he eminal payoff of a Eopean opion on he ndelying sock S wih sike pice K is max S K (4.) his means he holde will execise his igh only S K and hen his gain is S K. Ohewise if S K hen he holde will by he ndelying asse fom he make and he ale of he opion is zeo. We wold like o find a fomla fo picing a Eopean call opion wih sike pice K and maiy based on he model (3.5). Conside a coninos-ime economy whee inees aes ae sochasic and he pice of he Eopean call opion a ime nde he -fowad mease is CS K ; PE max S K S P ( ) max S K p S S ds whee E is he expecaion wih espec o he -fo- wad pobabiliy mease p is he coesponding con- diional densiy gien S and P is a zeo copon bond which is defined in Lemma. Wih a change in aiable X ln S C S ; K X X ln ln K X ln K KP p X X dx P max e K p X X dx P e K p X X dx X K KP p X X dx ln K (4.) Wih he fis inegand in (4.) being posiie and inegaing p o one. he fis inegand heefoe defines a new pobabiliy mease ha we denoe by C S ; K ln K X e q X X dx q below ln K d X K KP X K X K X P ln KP p X X X X e P ; P ; X e P ln KP X K X (4.3) whee hose pobabiliies in (4.3) ae calclaed nde he pobabiliy mease. he Eopean call opion fo log asse pice X ln S will be denoed by ˆ X C X ; e P X ; (4.4) e P P X ; whee ln K and P X ; := P X ; K. Noe ha we do no hae a closed fom solion fo hese = P e p X X dx pobabiliies. Howee hese pobabiliies ae elaed o chaaceisic fncions which hae closed fom solions as will be seen in Lemma 4. he following lemma shows ln K he elaionship beween P and P in he opion ale of X X e e p X X ln K X X (4.4). E e S Lemma 3 he fncions P and P in he opion al- KP p X X d es of (4.4) saisfy he PIDEs (4.5): X ln K and sbec o he bonday condiion a expiaion = p X X X P X x ; x. (4.6) e e dx ln K X E (e S ) Moeoe P saisfies he Eqaion (4.7) P P y A P e P x y ; P x ; kydy (4.5) P P P AP b x x a b ( P b P ) e b (4.7) P y P x y ; P x ; (e ) kydy x Copyigh SciRes.

6 S. PINKHAM E AL. 3 and sbec o he bonday condiion a expiaion = P x ; x (4.8) whee fo i = P i P i AP [ i ] e x P i P i P i P i P i (4.9) x x P i y P i x y ; P i( x ; ) e kydy x Noe ha x if x and ohewise x. Poof. See Appendix A. 5. he Closed-Fom Solion fo Eopean Call Opions Fo = he chaaceisic fncion fo P x ; wih espec o he aiable defined by ae iκ f ; : e d x P x ; (5.) wih a mins sign o accon fo he negaiiy of he mease dp. Noe ha f also saisfies simila PIDEs f A f x ; (5.) wih he especie bonday condiions i f x ; e d P x ; Since d P x ; d x xd he following lemma shows how o calclae he chaaceisic fncions fo P and P as hey appeaed in Lemma 3. Lemma 4 he fncions P and P can be calclaed by he inese Foie ansfomaions of he chaaceisic fncion i.e. i e f ; x P x ; Re d π i fo wih Re[.] denoing he eal componen of a complex nmbe. By leing he chaaceisic fncion f is gien by i ix e x d e. f x ; exp ix B C E whee b b b b b b i b i ix y y e e b i i k y dy ix y b i e i e k ydy b b b b e b B ln b b b b B e i i 3 3 4e e 3 4 i C e b 4b b b 4bb b b e E. b 4bb b b b e b b b b e b ln b b b b i 3 4i e ) 3 4 4i e i 4i Poof. See Appendix B. In smmay we hae s poed he following main heoem. Copyigh SciRes.

7 4 S. PINKHAM E AL. heoem 5 he ale of a Eopean call opion of SDE (3.5) is C S ; K SP X ; KP( ) P X ; whee P and P ae gien in Lemma 4 and P is gien in Lemma. 6. Acknowledgemens his eseach is (paially) sppoed by he Cene of Excellen in Mahemaics he commission on Highe Edcaion (CHE). Addess: 7 Rama VI Road Rachahewi Disic Bangkok hailand. 7. Refeences [] P. Ca and L. W ime Change Ley Pocesses and Opion Picing Jonal of Financial Economics Vol. 7 No. 4 pp doi:.6/s34-45x(3)7-5 [] R. C. Meon Opion Picing when Undelying Sock Rens ae Disconinos Jonal of Financial Economics Vol. 3 No pp doi:.6/34-45x(76)9- [3] D. Baes Jmp and Sochasic Volailiy: Exchange Rae Pocesses Implici in Deche Mak in Opion Reiew of Financial Sdies Vol. 9 No. 996 pp doi:.93/fs/9..69 [4] G. Yan and F. B. Hanson Opion Picing fo Sochasic Volailiy Jmp Diffsion Model wih Log Unifom Jmp Amplides Poceeding Ameican Conol Confeence Minneapolis 4-6 Jne 6 pp [5] Y. J. Kim Opion Picing nde Sochasic Inees aes: An Empiical Inesigaion Asia Pacific Financial Makes Vol. 9 No. pp doi:.3/a:55376 [6] D. Bigo and F. Mecio Inees Rae Models: heoy and Pacice nd Ediion Spinge Belin. [7] R. Bmmelhis Mahemaical Mehod fo Financial Engineeing Uniesiy of London 9. hp:// ds/lece.pdf [8] P.E. Ploe Sochasic Inegaion and Diffeenial Eqaion Sochasic Modeling and Applied Pobabiliy Vol. nd Ediion Spinge Belin 5. [9] N. Pial An Elemenay Inodcion o Sochasic Inees Rae Modeling Adance Seies on Saisical Science & Applied Pobabiliy Vol. Wold Scienific Singapoe 8. [] M. G. Kendall A. Sa and J. K. Od Adance heoy of Saisics Vol. Halsed Pess New Yok 987. Copyigh SciRes.

8 S. PINKHAM E AL. 5 Appendix A: Poof of Lemma 3 By Io s lemma Cˆ x follows he paial inego-diffeenial eqaion (PIDE) Cˆ LC D ˆ J ˆ LC (A.) whee ˆ ˆ D ˆ C C LC e x ˆ ˆ ˆ C C C Cˆ x Cˆ ˆ C x and J LCˆ ˆ ˆ ˆ C y Cx y Cx e kydy x whee k( y) is he Léy densiy. We plan o sbsie (4.4) ino (A.). Fisly we compe Cˆ x P P e e P P a b Cˆ x P P e P e P x x x Cˆ x P P e e P Cˆ x P P e e P Pb ˆ C x P P e P P e P x x x x Cˆ x P P e e P ˆ ˆ C x P P P C x P e e P e x x x P P e P b Pb. Cˆ x P P P P e e P b x x x x ; ˆ ; Cˆ x y C x y x e e P x y ; P x y ; P x ; e P ( ) P x y ; P x ;. Sbsie all ems aboe ino (A.) and sepaae i by assmed independen ems of P and P. his gies wo PIDEs fo he -fowad pobabiliy fo P x ; : i P P P e x P P P P x x P y P x y ; P x ; e kyd y. x y e P x y ; P x ; kyd y. P (A.) and sbec o he bonday condiion a he expiaion ime = accoding o (4.6). By sing he noaion in (4.9) hen (A.) becomes Eqaion (A.3) P P y AP e P x y ; P x ; kyd y. P : A P. (A.3) Copyigh SciRes.

9 6 S. PINKHAM E AL. Fo P ( x ; ): P P P P P P P b P x x x a b P e b P e b P y P x y ; P x ; e kyd y. x (A.4) and sbec o he bonday condiion a expiaion ime Again by sing he noaion (4.9) hen (A.4) becomes = accoding o (4.8). b P P P P a AP b P b x x P P e b( ): A P he poof is now compleed. (A.5) Copyigh SciRes.

10 S. PINKHAM E AL. 7 Appendix B: Poof of Lemma 4 o sole he chaaceisic fncion explicily leing be he ime-o-go we conece ha he fncion f is gien by f x ; (B.) exp ix B C E and he bonday condiion B C E. his conece explois he lineaiy of he coefficien in PIDEs (5.). Noe ha he chaaceisic fncion of f always exiss. In ode o sbsie (B.) ino (5.) fisly we compe f f B C E f if x f f f x C f E f f f E f C f f x ic f ie f ix e f ( x ; ) f f x f ( x y ; ) f ( x ; ) Sbsiing all he aboe ems ino (5.) afe cancelling he common faco of f we ge a simplified fom as follows: C ic ix y y ix E i E E i e i e e e k ydy + B e C C E By sepaaing he ode and odeing he emaining ems we can edce i o hee odinay diffeenial eqaions (ODEs) as follows: C() C () i (B.) E E ie ix y y i e ie k yd y (B.3) C B e C E. (B.4) I is clea fom (B.) and C() ha i C e (B.5) Le b b i ix y y e e and sbsie all em aboe ino (B.3). we ge b i i k y dy By mehod of aiable sepaaion we hae b b 4b b b b 4b b E b E E b b E Using paial facion on he lef hand side we ge de b b 4b b b b 4b b E b b b d Copyigh SciRes.

11 8 S. PINKHAM E AL. de b b E E( ) b b whee b 4b b. Inegaing boh sides we hae E ln E b b E b b Using bonday condiion E ( ) we ge b E ln Soling fo E ( ) we obain b E e bb b b e b whee b b b b. d (B.6) In ode o sole B ( ) explicily we sbsie C ( ) and E ( ) in (B.5) and (B.6) ino (B.4). i i i B ' e e e e e bb e b b e b Inegaing wih espec o and sing bonday condiion B ( ) hen we ge B i 3 3 i 4 e 4e e 3 b b b b e b ln b b b b he deails of he poof fo he chaaceisic fncion f ae simila o f. Hence we hae f x ; exp ix B C E whee B C and E ( ) ae as gien in his Lemma. We can hs ealae he chaaceisic fncion in close fom. Howee we ae ineesed in he pobabiliy P. hese can be ineed fom he chaaceisic fncions by pefoming he following inegaion P x ; iκ e f x. ; Re d i fo whee X ln S and ln K see Kendall e al. []. he poof is now complee. Copyigh SciRes.

General Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security

General Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security 1 Geneal Non-Abiage Model I. Paial Diffeenial Equaion fo Picing A. aded Undelying Secuiy 1. Dynamics of he Asse Given by: a. ds = µ (S, )d + σ (S, )dz b. he asse can be eihe a sock, o a cuency, an index,