PRICING AMERICAN PUT OPTION WITH DIVIDENDS ON VARIATIONAL INEQUALITY

Transcription

1 Joual of Mahemaical cieces: Aaces a Applicaios olume Pages 9-36 Aailable a hp://scieificaacescoi DOI: hp://oiog/0864/msaa_ PRICIG AMERICA PUT OPTIO ITH DIIDED O ARIATIOAL IEQUALITY XIAOFAG ZHOU a HAICHU TAG chool of Ifomaio ciece a Techology Jia Uiesiy Guagzhou 5063 P R Chia zhouf9908@63com Absac I his pape cosieig he aiaioal iequaliy moel fo Ameica pu opio wih iies ue Black-choles moel solig he aiaioal equaio by he fiie iffeece meho a he spliig meho i ime umeical epeimes hae eifie he effecieess of he algoihm Ioucio ice 973 he fis ime fo echage-ae fuues i he Uie aes opios make is eelopig ey apily Opio heoy was fis pu fowa by Black a choles i [] ow i has become a impoa ool fo fiacial isk maageme as we kow i [] he coe poblem is he opio picig poblem Thee is o aalyical soluio fo Ameica opios i mus use he umeical meho a has challeges Theefoe suyig he umeical mehos of he opio picig moel has impoa heoeical sigificace a pacical alue i [3] 00 Mahemaics ubec Classificaio: F8309 O63 Keywos a phases: Black-choles moel Ameica pu opio aiaioal iequaliy Receie Decembe cieific Aaces Publishes

2 0 XIAOFAG ZHOU a HAICHU TAG Opio picig moel umeical mehos icluig laice poi meho paial iffeeial equaio meho a Moe Calo sochasic simulaio meho Laice poi meho is simple he algoihm is easy o impleme bu he coegece spee is slow a he compuaio gows shaply as he umbe of oes of ime iecio iceases sochasic simulaio is o limie wih imesio bu is weakess is ha he slowe coegece spee i [4] A pese he eseach is base o paial iffeeial equaios of he umeical meho his meho uses he fiie iffeece meho i [5] combiig wih he bouay coiios he emiaio coiios a fially ge he pice of Ameica opios Base o esuls of omesic a foeig eseach i [6 7] his pape cosie he umeical soluio of he aiaioal iequaliy moel fo Ameica pu opio wih iies eep i suyig o he spliig meho i ime obaie he umeical algoihm wih iies a he epeimeal esuls show ha his meho is a ey effecie i solig he Ameica pu opio picig moel a he spliig meho is bee ha fiie iffeece meho The Ameica Pu Opio Picig Moel q T a K epese pice of he uelyig asse a ime ime cosa olailiy isk less iees ae i he bak iie ae epiaio ime a sike pice he Ameica pu opio ( ) is suiable fo he aiaioal iequaliy moel eek ( ) C i saisfies he followig equaliy: mi{ L ( K ) } 0 ( ) ( T ) ( K ) 0 < 0 ( ) ( 3)

3 PRICIG AMERICA PUT OPTIO ITH DIIDED fo L ( q) (4) Hee we iouce he fiie iffeece meho a he spliig meho i ime o sole he Ameica pu opio picig wih iies 3 The Fiie Diffeece Meho I oe o be able o sole he poblem ()-(3) o { : 0 < 0 T } we pefom he followig aiable asfomaio: o he moel of ()-(3) become fo l (3) K ( ) ( ) (3) K mi{ L0 ( e ) } 0 ( R 0 T ) ( 33) ( T ) ( e ) ( R) ( 34) 0 L q (35) cosuc he gi o ( R 0 T ) Q {( ) 0 Z} whee Z is he collecio of aual umbe T > 0 Defie fucio a each gi oe ( ) (36)

4 XIAOFAG ZHOU a HAICHU TAG ϕ ϕ e (37) e ake Pu hem io (33)-(34) ge he equaio a ϕ ϕ (39) (38) 0 0 mi q z Because 0 0 mi 0 mi > B A B A a ma 0 mi B A C B C A C I (38) ake ge { } 0 mi ϕ c (30) hee

5 PRICIG AMERICA PUT OPTIO ITH DIIDED 3 q a a c I (38) ake ge [ ] 0 mi ϕ c a so [ ] ma ϕ ϕ c a 0 z (3) Theoem 3 If a q The he iffeece scheme of (39) (3) is coege amely lim 0 whee is he liea eesio of is he iscosiy soluio of (33) (34)

6 4 XIAOFAG ZHOU a HAICHU TAG 4 The pliig Meho Cosie he fee bouay poblem of he Ameica pu opio moel wih iies { ( ) } is he soluio o (33)-(34) o : { < 0 T } wih he followig emial a bouay coiios: ( q) 0 ( ) ( 4) ( () ) K ( 4) ( () ) ( 43) ( T ) ( K ) ( 44) ( ) 0 ( ) ( 45) ( T ) K ( 46) Obiously if we pefom he followig aiable asfomaio: K () K K The we ca uiizaize amely assumig ha K we iie he 0 T io pas wih ieal [ ] 0 0 < < < T A each oe efie ( ) 与 T Make hem suiable fo a se of oiay iffeeial equaio of fee bouay poblems by (4)-(46) o isceizaio amely seek { } ( 0 ) i saisfies he followig equaliy:

7 PRICIG AMERICA PUT OPTIO ITH DIIDED 5 ( q) 0 ( < ) ( 47) ( ) ( 48) ( ) ( 49) 0 ( ) ( 40) ( ) ( 4) ( 4) Hee 0 The followig is specific algoihm: () Because ( ) ( 0 < ) o we ge he alue of whe () Fom he iucio pocess we kow a < efie < 0 < 0 Obiously C [ ) (3) To sole he followig fee bouay poblem o { < } L q ( < ) ( 43) ( 44) ( ) ( 45) 0 ( ) ( 46)

8 6 XIAOFAG ZHOU a HAICHU TAG e ge { } (4) To sole he iffeeial equaio of he fee bouay poblem (43)-(46) hee ae may ways o choose Cosieig he paiculaiy of his equaio ca ge a epeseaio of he soluio o Equaio (43) e will iouce how o use meho iecly o solig a Fisly sole a Equaio ( 43) is a homogeeous seco oe oiay iffeeial equaio Fis sole he geeal fom of homogeeous equaio make Plug i (43) we ge K ( ) ( ) ( q) 0 (47) Equaio (47) is calle he chaaceisic equaio of oiay iffeeial equaio (43) i has wo oos: ± ρ γ ± Hee γ q ρ q o > 0 > Theefoe coespoig soluio of (43) has he followig fom: ( )

9 PRICIG AMERICA PUT OPTIO ITH DIIDED 7 fom he aue of he bes implemeaio bouay Theefoe < The igh pa of (43) ca be ewie as he fom of piecewise polyomial L 0 ( < ) ( 48) L ( ) ( < ) ( 49) coiuous a ( 40) a suiable fo bouay coiios (44)-(46) Because he ohomogeeous equaio (49) hae a special soluio ( q) o he soluio of Equaio (48)-(49) is Hee () () ( ) ( 4) ( q) ( ) ( 4) i ( i m ) a ae o be eemie hey ae mae up m of bouay coiios (44)-(46) a coecio coiios (40) Fom (46) we kow 0 (43) Fom (40) we ge () () () q (44) () () () q (45) Fom he fee bouay coiios of (44) a (45) we kow () () 0 (46) q

10 8 XIAOFAG ZHOU a HAICHU TAG () () (47) q Fom (44) a (45) we ge amely ( ) () q q () ( ) ( ) ρ q q (48) Fo ( 44) ( 47) we ge he equaios which is suiable fo () ρ 0 (49) ( q) ( q) o ca be wok ou Easy o kow 0 < pu < i (47) ge () () (430) ( q) () () Pu a i (44) we ge { }; o we will wok ou ( 0 ) () () q ( ) ( ) Amog hem he efiiio of (43) a (49) a i (48) (430)

11 PRICIG AMERICA PUT OPTIO ITH DIIDED 9 Lemma 4 Cosieig ohomogeeous oiay iffeeial equaios; q L (43) is he sigle oo of chaaceisic equaio (47) amely 0 K (43) The Equaio (43) hae a special soluio l K ow seek { } They ae suiable fo he followig equaios: () () a coiuous ae 436 a coiuous ae q L L L Equaio (435) has a special oo q (44)

12 30 XIAOFAG ZHOU a HAICHU TAG The Equaios (433) a (434) hae special oos as follows: () () l K ( ) l ( ) K (44) ( ) ( q) l (443) K 3 o o { < } he geeal oos of (433)-(435) ca be wie as () ( ) ( 444) ( ) ( 445) 3 3 ( ) ( 446) 3 Use coiios (436) (437) (438) (439) a (440) o calculae ( m) i ( m 3 i ) a Fom (440) we ge ( 3) 0 (447) Fom (436) we ge () () ( 3) (448) 3 () () ( 3) (449) 3 Because () 3() 0 ( ) ( q) so [ () 3()] ρ ( ) ( ) (450) q

13 PRICIG AMERICA PUT OPTIO ITH DIIDED 3 Fom (437) we ge () (45) () o () [ ] [ ] ρ (45) Fom (438) a (439) we ge () (453) Because q q hus () ρ q q (454) Plug (450) (45) io (454) we go suiable fo asceeal equaio < Fom (453) we ca ge ow we kow we ca ife fom (45) pu i i (448) we ca ge 3 ow we hae calculae all of he ueemie coefficies 3 m i m i a The ifeece pocess ae as follows:

14 3 XIAOFAG ZHOU a HAICHU TAG ( 3) ( 3) Usig backwa iucio pocess if C [ 0 ) a ( 0) ae kow whe ca be iie io he followig ieals: [ ] [ ] [ ) 0 he M 0 O [ ) is piecewise fucio Amog hem M [ ] Hee ( l i ) ( l i ) ( M c e f ) i i i ( c ) i i e f ae cosa ae oos of (47) so i oe o fi a whe we shoul ewie (43)-(46) as follows: i L L L L M ( ) ( 455) M M M ( ) ( 456) ( ) ( 457) ( 458)

15 PRICIG AMERICA PUT OPTIO ITH DIIDED 33 ae coiuous a (459) ae coiuous a (460) ( ) (46) (46) ( ) 0 (463) () ( ) ( < ) Hee i ( i ) ae special oos of (455)-(458) whose ( igh sie has he fom of Mi we ca fi m) ( m ; ) a fom (459) (460) (46) (46) a (463) whe he sole hem i u ( ) ( ) 5 umeical Eamples Cosie umeical simulaio of Ameica pu opio picig poblem i [8] i ()-(3) se 0 K 00 T a q ake he followig wo kis of siuaios: ) 0 05 q 005; ) 0 q 00 I oe o eify he umeical esuls of iffee algoihm umeical esuls ae show i Figue a Figue Hee epeses ime epeses sock pice; P epeses he alue of opio Coas esul of he fiie iffeece meho a he spliig meho is gie i Figue Figue shows he 3D images abou he alue of opio The umeical esuls shows ha spliig meho ca mach he opio pice moe accuaely

16 34 XIAOFAG ZHOU a HAICHU TAG Figue The bes implemeaio boe calculae by fiie iffeece meho a spliig meho

17 PRICIG AMERICA PUT OPTIO ITH DIIDED 35 Figue Thee-imesio images abou he alue of opio

18 36 XIAOFAG ZHOU a HAICHU TAG Refeeces [] F Black a M choles The picig of opios a copoae liabiliies J Plli Eco 8(3) (973) [] L Jiag Mahemaical Moelig a Mehos of Opio Picig Beiig: Highe Eucaio Pess (008) 7-00 [3] Zheg a J F Yi Moulus-base successie oe elaaio meho fo picig Ameica opios Appl Mah Ifomaics 3(5) (03) [4] P Glassema Moe Calo Mehos i Fiacial Egieeig ew Yok: pige- elag (004) 0-6 [5] Zheg a J F Yi O he coegece of poece iagula ecomposiio mehos fo picig Ameica opios wih sochasic Appl Mah Compu 3(4) (03) 4-4 [6] M Bea a E chwaz The aluaio of Ameica pu opios J Fia 3(5) (977) [7] H Ha a X A u Fas umeial meho fo he Black-choles equaio of Ameica opios IAM J ume Aal 4(6) (004) -53 [8] Z F He The Applicaio of MATLAB i Mahemaics Tsighua Uiesiy Pess (0) g