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

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1 Jonal of Mahemacal Fnance 98-8 do:436/mf33 Pblshed Onlne Noembe (hp://wwwscrpog/onal/mf) Eopean Opon Pcng fo a Sochasc Volaly Léy Model wh Sochasc Inees Raes Sasa Pnkham Paoe Saayaham School of Mahemacs Inse of Scence Sanaee Unesy of echnology Nakhon Rachasma haland E-mal: sasa@mahsach paoe@sach Receed Ags 7 ; esed Sepembe 9 ; acceped Sepembe 8 Absac We pesen a Eopean opon pcng when he ndelyng asse pce dynamcs s goened by a lnea combnaon of he me-change Léy pocess and a sochasc nees ae whch follows he Vascek pocess We oban an explc fomla fo he Eopean call opon n em of he chaacesc fncon of he al pobables Keywods: me-change Léy Pocess Sochasc Inees Rae Vascek Pocess Fowad Mease Opon Pcng Inodcon Le F P be a pobably space A sochasc pocess L s a Léy pocess f has ndependen and saonay ncemens and has a sochascally connos sample pah e fo any lm P L L he sm- h h ples possble Léy pocesses ae he sandad Bownan moon W Posson pocess N and compond Posson N pocess Y whee N s Posson pocess wh nensy and Y ae d andom aables Of cose we can bld a new Léy pocess fom known ones by sng he echnqe of lnea ansfomaon Fo example N he mp dffson pocess W Y whee ae consans s a Léy pocess whch comes fom a lnea ansfomaon of wo ndependen Léy pocesses e a Bownan moon wh df and a compond Posson pocess Assme ha a sk-neal pobably mease exss and all pocesses n secon wll be consdeed nde hs sk-neal mease In he Black-Scholes model he pce of a sky asse S nde a sk-neal mease and wh non ddend paymen follows S SexpL SexpW () whee s a sk-fee nees aes s a o- laly coeffcen of he sock pce Insead of modelng he log ens L W wh a nomal dsbon We now eplace wh a moe sophscaed pocess L whch s a Léy pocess of he fom L W J () whee J and denoes a pe Léy mp componen (e a Léy pocess wh no Bownan moon pa) and s conexy adsmen We assme ha he pocesses W and J ae ndependen o ncopoae he olaley effec o he model () we follow he echnqe of Ca and W [] by sbodnang a pa of a sandad Bownan moon W and a pa of mp Léy pocess J by he me negal of a mean eeng Cox Ingesoll Ross (CIR) pocess ds s whee follows he CIR pocess d d d W (3) Hee W s a sandad Bownan moon whch coesponds o he pocess he consan s he ae a whch he pocess ees owad s long em mean and s he olaly coeffcen of he pocess Copygh ScRes

2 S PINKHAM E AL 99 Hence he model () has been changed o L W J (4) and hs new pocess s called a sochasc olaly Ley pocess One can nepe as he sochasc clock pocess wh acy ae pocess By eplacng L n () wh L we oban a model of an ndelyng asse nde he sk-neal mease wh sochasc olaly as follows: S S expw J (5) In hs pape we shall consde he poblem of fndng a fomla fo Eopean call opons based on he ndelyng asse model (5) fo whch he consan nees aes s eplaced by he sochasc nees aes and J s compond Posson pocess e he model nde o consdeaon s gen by S S exp W J (6) Hee we assme ha follows he Vascek pocess d d dw (7) W s a sandad Bownan moon wh espec o he pocess and dw d d W W dw he consan s he ae a whch he nees ae ees owad s long em mean s he olaly coeffcen of he nees ae pocess (7) he consan s a speed eeson Leae Reews Many fnancal engneeng sdes hae been ndeaken o modfy and mpoe he Black-Scholes model Fo example he mp dffson models of Meon [] he sochasc Volaly mp dffson model of Baes [3] and Yan and Hanson [4] Fhemoe he me change Léy models poposed by Ca and W [] he poblem of opon pcng nde sochasc nees aes has been nesgaed fo along me Km [5] consced he opon pcng fomla based on Black-Scholes model nde seeal sochasc nees ae pocesses e Vascek CIR Ho-Lee ype He fond ha by ncpoang sochasc nees aes no he Black-Scholes model fo a sho may opon does no conbe o mpoemen n he pefomance of he ognal Black- Scholes pcng fomla Bgo and Meco [6] menon ha he sochasc feae of nees aes has a songe mpac on he opon pce when pcng fo a long may opon Ca and W [] conne hs sdy by gng he opon pcng fomla based on a me-changed Léy pocess model B hey sll se consan nees aes n he model In hs pape we ge an analyss on he opon pcng model based on a me-changed Léy pocess wh sochasc nees aes he es of he pape s oganzed as follows he dynamcs nde he fowad mease s descbed n Secon 3 he opon pcng fomla s gen n Secon 4 Fnally he close fom solon fo a Eopean call opon n ems of he chaacesc fncon s gen n Secon 5 3 he Ddynamcs nde he Fowad Mease We begn by gng a bef eew of he defnon of a coelaed Bownan moon and some of s popees (fo moe deals one see Bmmelhs [7]) Recallng n ha a sandad Bownan moon n R s a sochasc pocess Z whose ale a me s smply a eco of n ndependen Bownan moons a Z Z Zn We se Z nsead of W snce we wold lke o esee he lae fo he moe geneal case of coelaed Bownan moon whch wll be defned as follows: be a (consan) pose symmec Le n max sasfyng and By Cholesky s decomposon heoem one can fnd an ppe angl n n max h sch ha whee Η s he anspose of he max Η Le Z Z Zn be a sandad Bownan moon as nodced aboe we defne a new eco-aled pocess W W Wn by W Z o n em of componens n W hz n he pocess W s called a coelaed Bownan moon wh a (consan) coelaon max Each componen pocess W s self a sandad Bownan moon Noe ha f Id (he deny max) hen W s a sandad Bownan moon Fo example f we le a symmec max (3) hen has a Cholesky decomposon of he fom HH whee H s an ppe angla max of he fom Copygh ScRes

3 S PINKHAM E AL H Le Z Z Z moons hen W W W Z be hee ndependen Bownan W defned by W Z o n 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 dynamc gen by Y ds S d dw S e d N m d d d W d d d W whee e Y m E (33) dwdw dwdw and dwdw d We can e-we he dynamc (33) n ems of hee ndependen Bownan moons Z Z Z follows (3) we ge ds S md dz dz (34) Y S e d N d d d Z (35) d d d Z (36) hs decomposon makes ease o pefom a mease ansfomaon In fac fo any fxed may le s denoe by he -fowad mease e he pobably mease ha s defned by he Radon- Nkodym deae exp d d (37) d P Hee P s he pce a me of a zeo-copon bond wh may and s defned as e ds s P E F (38) Nex Consde a connos-me economy whee nees aes ae sochasc and sasfy (35) Snce he SDE (35) sasfes all he necessay condons of heoem 3 see Poe [8] hen he solon of (35) has he Mako popey As a conseqence he zeo copon bond pce a me nde he mease n (38) sasfes P E exp sds (39) Noe ha P depends on only nsead of dependng on all nfomaon aalable n F p o me As sch becomes a fncon F of P F meanng ha he pcng poblem can now be fomlaed as a seach fo he fncon F Lemma he pce of a zeo copon bond can be deed by compng he expecaon (39) We oban P exp a b (3) whee b e 3 a e e 3 Poof See Pal [9] (pp 38-39) Lemma he pocess followng he dynamcs n (35) can be wen n he fom whee he pocess x w fo each (3) x sasfes dx d d x Z x (3) Moeoe he fncon w() s deemnsc and well defned n he me neal [] whch sasfed e e w In pacla w (33) Poof o sole he solon of SDE (35) g e and sng Io s Lemma Le g g g dg d d d hen de e de ddz (34) = e d e d Z Inegaed on boh sde he aboe eqaon fom o whee and smplfed one ge e e e dz By sng he defnon of w fom (33) Copygh ScRes

4 S PINKHAM E AL d w e Z (35) e e w Noe ha he solon of (3) s whee e e d e d (36) x x Z Z Hence w x fo each he poof s now complee Nex we shall calclae he Radon-Nkodym deae as appea n (37) By Lemma and we hae x w and P Sbsng and P no (37) we hae x w d d exp d exp a b (37) exp x d e d Sochasc negaon by pas mples ha x d x d x d x (38) By sbsng he expesson fo dx fom (3) dx (39) x d d Z Moeoe by sbsng he expesson fo x fom (36) he fs negal on he gh hand sde of (39) becomes x d s d o e Z d Usng negal by pas we hae (Eqaon 3) Sbsng (3) no (39) we oban dx d e Z (3) Hence x d e d Z (3) Sbsng (3) no (37) once ge d d exp e dz e d (33) he Gsano heoem hen mples ha he hee poc esses Z Z and Z defned by d d e Z d Z (34) dz d Z dz dz ae hee ndependen Bownan moons nde he meas- e heefoe he dynamcs of and S nde ae gen by Y d ds S d dz dz m S e N d e dd Z d d d Z (35) 4 he Pcng of a Eopean Call Opon on he Gen Asse Le S be he pce of a fnancal asse modeled as a sochasc pocess on a fleed pobably space FF F s sally aken o be he pce hsoy p o me All pocesses n hs secon wll be defned n hs space We denoe C he pce a me of a Eopean call opon on he cen pce of an ndelyng asse S wh ske pce K and expaon me 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) Copygh ScRes

5 S PINKHAM E AL he emnal payoff of a Eopean opon on he ndelyng sock S wh ske pce K s max S K (4) hs means he holde wll execse hs gh only S K and hen hs gan s S K Ohewse f S K hen he holde wll by he ndelyng asse fom he make and he ale of he opon s zeo We wold lke o fnd a fomla fo pcng a Eopean call opon wh ske pce K and may based on he model (35) Consde a connos-me economy whee nees aes ae sochasc and he pce of he Eopean call opon a me nde he -fowad mease s CS K ; PE max S K S P ( ) max S K p S S ds whee E s he expecaon wh espec o he -fo- wad pobably mease p s he coespondng con- donal densy gen S and P s a zeo copon bond whch s defned n Lemma Wh a change n aable X ln S C S ; K X P max e K p X X dx X ln P e K p X X dx ln K X ln K = P e p X X dx KP p X X dx ln K X e e p X X X e X X ln K E e S d ln K p X X X e dx X X K KP p X X X X ln K E (e S ) KP p X X d X ln K (4) Wh he fs negand n (4) beng pose and negang p o one he fs negand heefoe defnes a new pobably mease ha we denoe by C S ; K ln K X e q X X dx q below d ln K 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 (43) whee hose pobables n (43) ae calclaed nde he pobably mease he Eopean call opon fo log asse pce X ln S wll be denoed by ˆ X C X ; e P X ; (44) e P P X ; whee ln K and P X ; := P X ; K Noe ha we do no hae a closed fom solon fo hese pobables Howee hese pobables ae elaed o chaacesc fncons whch hae closed fom solons as wll be seen n Lemma 4 he followng lemma shows he elaonshp beween P and P n he opon ale of (44) Lemma 3 he fncons P and P n he opon ales of (44) sasfy he PIDEs (45): and sbec o he bonday condon a expaon = P x ; x (46) Moeoe P sasfes he Eqaon (47) P P y AP e P x y ; P x ; kydy (45) P P P AP b x x a b ( ) P b P e b (47) P y P x y ; P x ; (e ) kydy x Copygh ScRes

6 S PINKHAM E AL 3 and sbec o he bonday condon a expaon = P x ; x (48) whee fo = P P AP [ ] e x P P P P P (49) x x P y P x y ; P ( x ; ) e kydy x Noe ha x f x and ohewse x Poof See Appendx A 5 he Closed-Fom Solon fo Eopean Call Opons Fo = he chaacesc fncon fo P x ; wh espec o he aable ae defned by κ f ; : e d x P x ; (5) wh a mns sgn o accon fo he negay of he mease dp Noe ha f also sasfes smla PIDEs f A f x ; (5) wh he espece bonday condons f x ; e d P x ; x e x d e Snce P x d x d ; x d he followng lemma shows how o calclae he cha- acesc fncons fo P and P as hey appeaed n Lemma 3 Lemma 4 he fncons P and P can be calclaed by he nese Foe ansfomaons of he chaacesc fncon e e f ; x P x ; Re d π fo wh Re[] denong he eal componen of a complex nmbe By leng he chaacesc fncon f s gen by f x ; exp x B C E whee b b b b b b b x y y b e e kydy x y b e e k ydy b b b b e b B ln b b b b 3 e 3 4e e 3 4 C e b 4bb b 4bb b b e E b 4bb b b b e b b b b e b B ln b b b b 3 4 e ) 3 4e 4 4 Poof See Appendx B In smmay we hae s poed he followng man heoem Copygh ScRes

7 4 S PINKHAM E AL heoem 5 he ale of a Eopean call opon of SDE (35) s C S ; K SP X ; KP ( ) P X ; whee P and P ae gen n Lemma 4 and P s gen n Lemma 6 Acknowledgemens hs eseach s (paally) sppoed by he Cene of Excellen n Mahemacs he commsson on Hghe Edcaon (CHE) Addess: 7 Rama VI Road Rachahew Dsc Bangkok haland 7 Refeences [] P Ca and L W me Change Ley Pocesses and Opon Pcng Jonal of Fnancal Economcs Vol 7 No 4 pp 3-4 do:6/s34-45x(3)7-5 [] R C Meon Opon Pcng when Undelyng Sock Rens ae Dsconnos Jonal of Fnancal Economcs Vol 3 No pp 5-44 do:6/34-45x(76)9- [3] D Baes Jmp and Sochasc Volaly: Exchange Rae Pocesses Implc n Deche Mak n Opon Reew of Fnancal Sdes Vol 9 No 996 pp 69-7 do:93/fs/969 [4] G Yan and F B Hanson Opon Pcng fo Sochasc Volaly Jmp Dffson Model wh Log Unfom Jmp Ampldes Poceedng Amecan Conol Confeence Mnneapols 4-6 Jne 6 pp [5] Y J Km Opon Pcng nde Sochasc Inees aes: An Empcal Inesgaon Asa Pacfc Fnancal Makes Vol 9 No pp 3-44 do:3/a:55376 [6] D Bgo and F Meco Inees Rae Models: heoy and Pacce nd Edon Spnge Beln [7] R Bmmelhs Mahemacal Mehod fo Fnancal Engneeng Unesy of London 9 hp://wwwemsbbkack/fo_sdens/msc/mah_meho ds/lecepdf [8] PE Ploe Sochasc Inegaon and Dffeenal Eqaon Sochasc Modelng and Appled Pobably Vol nd Edon Spnge Beln 5 [9] N Pal An Elemenay Inodcon o Sochasc Inees Rae Modelng Adance Sees on Sascal Scence & Appled Pobably Vol Wold Scenfc Sngapoe 8 [] M G Kendall A Sa and J K Od Adance heoy of Sascs Vol Halsed Pess New Yok 987 Copygh ScRes

8 S PINKHAM E AL 5 Appendx A: Poof of Lemma 3 By Io s lemma Cˆ x follows he paal nego-dffeenal eqaon (PIDE) Cˆ LC D ˆ J ˆ LC (A) whee ˆ ˆ D ˆ C C LC e x ˆ C ˆ ˆ ˆ C C C x Cˆ ˆ C x and J LCˆ ˆ ˆ ˆ C y Cx y Cx e kydy x whee k( y) s he Léy densy We plan o sbse (44) no (A) Fsly 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 ep 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 P x y P x y P x x e e ; ; ; e P ( ) P x y ; P x ; Sbse all ems aboe no (A) and sepaae by assmed ndependen ems of P and P hs ges wo PIDEs fo he -fowad pobably fo P x ; : P P P e x P 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 (A) and sbec o he bonday condon a he expaon me = accodng o (46) By sng he noaon n (49) hen (A) becomes Eqaon (A3) P P y AP e P x y ; P x ; kyd y P : A P (A3) Copygh ScRes

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 (A4) and sbec o he bonday condon a expaon me Agan by sng he noaon (49) hen (A4) becomes = accodng o (48) b P P P P a AP b P b x x P P e b( ): A P he poof s now compleed (A5) Copygh ScRes

10 S PINKHAM E AL 7 Appendx B: Poof of Lemma 4 o sole he chaacesc fncon explcly leng be he me-o-go we conece ha he fncon f s gen by f x ; (B) exp x B C E and he bonday condon B C E hs conece explos he lneay of he coeffcen n PIDEs (5) Noe ha he chaacesc fncon of f always exss In ode o sbse (B) no (5) fsly we compe f f B C E f f x f f f C f E f f x f E f C f f x C f E f x e f ( x ; ) f f x f ( x y ; ) f ( x ; ) Sbsng all he aboe ems no (5) afe cancellng he common faco of f we ge a smplfed fom as follows: C C x y y x E E E e e e e k ydy + B e C C E By sepaang he ode and odeng he emanng ems we can edce o hee odnay dffeenal eqaons (ODEs) as follows: C() C () (B) E E E x y y e e k yd y (B3) B e C E C (B4) I s clea fom (B) and C() ha Le C e (B5) b b x y y b e e k y dy and sbse all em aboe no (B3) we ge b b 4bb b b 4bb E b E E b b By mehod of aable sepaaon we hae Usng paal facon on he lef hand sde we ge de bd b b 4bb b b 4bb E E b b Copygh ScRes

11 8 S PINKHAM E AL de b b E E( ) b b whee b 4bb Inegang boh sdes we hae b E b ln E b E b Usng bonday condon E ( ) we ge b E ln b Solng fo E ( ) we oban e bb E b b e b d (B6) whee b b b b In ode o sole B ( ) explcly we sbse C ( ) and E ( ) n (B5) and (B6) no (B4) B ' e e e bb e e e b b e b Inegang wh espec o and sng bonday condon B ( ) hen we ge B e 4e e 3 b b b b e b ln b b b b he deals of he poof fo he chaacesc fncon f ae smla o f Hence we hae f x ; exp x B CE whee B C and E ( ) ae as gen n hs Lem- ma We can hs ealae he chaacesc fncon n close fom Howee we ae neesed n he pobably P hese can be need fom he chaacesc fncons by pefomng he followng negaon P x ; κ e f x ; Re d fo whee X ln S and ln K see Kendall e al [] he poof s now complee Copygh ScRes