Option Pricing Model With Continuous Dividends

Transcription

1 Avacs i Naural Scic Vol. 8 No. 3 5 pp. -5 DOI:.3968/736 ISSN [PRIN] ISSN [ONLINE] Opio Pricig Mol Wih Coiuous Divi ZHENG Yigchu [a] ; YANG Yufg [a] ; ZHANG Shougag [a] [a] School of Scic Xi a vrsiy of Scic a chology Xi a Chia. Corrspoig auhor. Suppor by Naioal Naural Scic Fouaio of Chia ( ; Scic a chology Fouaio of Shaaxi Provic of Chia (3XJXX-4. Rciv 9 May 5; accp 8 July 5 Publish oli 6 Spmbr 5 Absrac his papr iscusss h problm of pricig o Europa opios i jump-iffusio mol by marigal mho. W assumig jump procss ar mor comm h Possio procss a ki of oxplosiv couig procss. Supposig ha h ivi for ach shar of h scuriy is pai coiuously i im a a ra qual o a fix fracio of h pric of h scuriy. By chagig h basic assumpio of R.C.Mro opio pricig mol o h assumpio. I is sablish ha h bhavior mol of h sock pricig procss is jump-iffusio procss. Wih risk-ural marigal masur pricig formula a pucall pariy for Europa opios wih coiuous ivi ar obai by sochasic aalysis mho. h rsuls of Margrab ar graliz. y wor: Coiuous ivi; Jump-iffusio; Opio pricig; Cou procss Zhg Y. C. Yag Y. F. & Zhag S. G. (5. Opio Pricig Mol Wih Coiuous Divi. Avacs i Naural Scic 8(3-5. Availabl from: hp:// DOI: hp://x.oi.org/.3968/736 INRODUCION Opio pricig hory is always o of h krl problms o fiacial mahmaics. oghr wih h capial ass pricig hory h porfolio slcio hory h ffciv hory of mark a acig issu i is rgar as o of h fiv hory mouls i mor fiac. May scholars hav o a gra al of rsarchs o opio pricig hory a obai may rsuls which ar isruciv i fiacial pracic. Howvr h apparac of impora iformaio will caus h sock pric o a ki of o coiual jumps. A mass of fiac pracicial has iica ha hr is a srious warp bw h hypohsis of Black-Schols mol (Black & Schols 973 for h urlyig ass pric a h ralisic marks. hrfor may scholars pu forwar may w ki of opio pricig mols (Ball & Roma 993; Harol & ushr ; Gill & Wog ; Cai & Mao ; ou by rlaxig som assumig coiios of Black-Schols mol. Opio pricig hory wih jump-iffusio is o of hm. Bu w show ha ral aa cao always b fi by a gomric Browia moio mol a ha mor gral mols may o b cosir. h apparac of impora iformaio will caus h sock pric o a ki of o coiual jumps (Yag Zhag & Xia 3; Rigr ; Yag & Hao 3; Aglo 9. Wh marks ar compl h xisc of opimal sragis ca b fou Mro (97 Jablac a Poir (99. Follmr a Lukr ( iscuss smimarigal mol Pham ( iscuss coiuous marks mol Nakao (4 iscuss jump-iffusio mo. Mro (976 sablish famous jump-iffusio mol wh jump procss is Possio procss a iscuss h impac of h ivi o h opio h Black-Schols formula was x. Roll a Gsk also pu forwar h pricig mol of h Amrica call opio wih ivi. h Black-Schols parial iffrial quaio was moifi afr hy cosir payig a ivi a h quaio was solv. I a coiuous sup whr h voluio of a sigl sock is moll by gomric Browia moio Black a Schols riv a clos-form soluio for h valu of Europa-syl call a pu opios by prsig a sragy ha uplicas is payoff hrough coiuous Copyrigh Caaia Rsarch & Dvlopm Cr of Scics a Culurs

2 Opio Pricig Mol Wih Coiuous Divi raig i h sock a h bo. O of h rawbacks of usig gomric Browia moio as a mol for a scuriy s pric ovr im is ha i os o allow for h possibiliy of a iscoiuous pric jump i ihr h up or ow ircio.bcaus such jumps o occur i pracic i is avaagous o cosir a mol for pric voluio ha suprimposs raom jumps o a gomric aa-is ha h pric crass by xacly h ivi pai. L (ΩFP (F b a probabiliy spac a {W ( } b a saar Wir procss giv o a probabiliy spac (ΩFP (F h mark is buil wih a bo B( a a risky ass S(. W suppos ha B( is h soluio of h quaio Browia moio. B( = r(b( B( =. ( Opios ar xampls of xchag-ra rivaiv scuriis ha isscuriis whos valu p o h S(saisfis h sochasic iffrial quaio S( = S(-[(μ( + σ(w ( + (U-M(] S( =s ( prics of ohr mor basic scuriis such as socks or bo. h opio pric is h oly variabl ha chags wih h mark supply a ma which ircly affc h profi a loss of h buyrs a sllrs ha is h cor issu of h opios raig. I his papr W cosir ha pric of urlyig ass pric obys jump-iffusio procss bcaus of h raliy h sock pric jumps o o cssarily oby h Poisso procss jump procss graliz coforms o h acual siuaio of sock pric movm. W Esablish h opio-pricig mol wih coiuous ivi. Pricig formula of Europa opio is also giv. h rsuls of xisig ar graliz. whr r( is risk-fr irs ra μ( is xpc sock rurs σ( is volailiy. M( = N(-λ(s is h compsa marigal of oxplosiv couig procss {N } wih isiy paramr λ(.w assum ha h filraio (F is gra by h {W ( } a marigal {M( }. L us cosir h cas wh h ivi yilrah ha h ivi payoffsis assum o b kow.mor spcifically w assum ha h sock coiuously pays ivi a som fix ra.followig h classic papr by Samulso a Mro (969 w assum ha h ffciv ivi ra is proporioal o h lvl of h sock pric. Alhough his is rahr impracical as a ralisic ivi policy associa wih a paricular. CONINUOUS DIVIDENS MODELS For isacif h sock s pric is prsly h i h x im uis h ivi paym pr shar of sock ow will b approximaly ρs wh is small.o bgiw a mol for h voluio of h pric of h scuriy ovr im. O way o obai a rasoabl mol is o suppos ha all ivi ar rivs i h purchas of aiioal shars of h sock.husw sock Samulso a Mro s mol fis h cas of a sock ix opyio rsoably wll.h ivi payms shoul b us i fullihr o purchas aiioal shars of sock or o ivs i risk fr bo. Cosqulya raig sragy is sai o b slf-fiacig wh is walh procss saisfis fiiio. Dfiiio (Yua 8 A sragy {a(b(}is call slf-fiacig if walh procss V( = a(s( + b(b( is saisfi woul b coiuously aig aiioal shars a h V( = a(s( + b(b( + ρ(a(s(. (3 ra ρ ims h umbr of shars w prsly ow. Cosqulyour umbr of shars i growig by a coiuously compou ra ρ.hrforif w purchas h coiuous ivi ra is ρ( i follows from (((3 ha V( = {a(s((u( + ρ( + b(r(b(}+ a sigl shar a im a im w woul hav ρ shars a(s(σ(w ( + a(s((u-m(. (4 wih a oal mark valu of ρ S. I sms rasoabl o L r ( s s V = V r ( s s S = S w ca prov suppos ha ρ S follows a gomric browia moio. I is usual o suppos haa h mom h ivi h followig proposiio. is pai h pric of a shar isaaously crass by Proposiio h followig ar quival h amou of h ivi.if o assums ha h pric (A sragy {a( b(}is call slf-fiacig. vr rops by a las h amou of h ivih ( Walh procss saisfi buyig immialy bfor a sllig immialy afr V ( = a(s ((σ(w( + (U-M( whr h paym of h ivi woul rsul i a arbirag; u + ρ r W = W + θ θ =. hc hr mus b som possibiliy of a rop i pric σ of a las h amou of h ivi a h usual assumpio-which is roughly i agrm wih acual Proof Applyig Io s lmma yil (Yag & Luo 6 w hav V = S a ( u + ρ r + σ ( W + ( U M (5 l W = W + θ(s θ( = u(+ρ(-r( σ( (5 quivally V = a S σ W + ( U M. ( W fi i covi o irouc a auxiliary procss ρ S = S whos yamics ar giv by h sochasic iffrial quaio. Copyrigh Caaia Rsarch & Dvlopm Cr of Scics a Culurs

3 ZHENG Yigchu; YANG Yufg; ZHANG Shougag (5. Avacs i Naural Scic 8(3-5 ρ Proposiio L S = S. W hav {S (} is saisfi ( σ S S W ( U M. Proof For S ρ = S a s S = S usig Io s lmma yil w hav = +. (6 r s ρ s ρ s s S = S + ρ S (7 r s r ( s S = S r S. (8 oghr wih ( (7 a (8 w hav S = S ( u + ρ r + σ W + ( U M (9 l W = W + θ(s θ( = u(+ρ(-r( (9 quivally σ( ( σ S = S W + ( U M. Applyig Dolas-Da xpoial formula (Duffi 996 sochasic iffrial quaio (6 quivally N S = S xp{ σ + ( E( U λ + σ W }. i= N so S = s xp{ [ r ρ σ ] + ( E( U λ + σ W }. whr i=. ( P Proposiio 3 L = xp θ W θ P bouw hav s V = V is P marigal.i follows ha V ( = E P (V ( F.Fially θ a E(U b bouh slf-fiacig r ( s V = EP V F. r ( s walh procss is V = EP V F. P is riskural marigal masur.. OPION PRICING FORMULA WIH Proof Applyig Girsaov horm W( = W ( + CONINUOUS DIVIDEN θ(s is saar Wir procss ur h marigal Proposiio 4 Assum ha h yamics of a bo B( a a PM(( is P marigal. risky ass S( ar giv by ( ( mauriy a xrcis For bcaus σ( is igrabl fucioa E(U is pric. h h pric of Europa call opio saisfy r = Φ ρ + λ i Φ = i= C( S P ( E[ s ( xp{ [ ( E( U ] } U ( ] s i= l + ( E( U r λ + ρ σ + = =. σ s s σ r s Proof Sic s V = V is h marigal ur risk-ural marigal masur h r s r ( s s C S E S + ( = [ ( F ] r s s r s s { S } { S } = E S I F E I F ( 3 Copyrigh Caaia Rsarch & Dvlopm Cr of Scics a Culurs

4 Opio Pricig Mol Wih Coiuous Divi S s xp{ [ r s ] ( E( U i s = s W }. r r E S I{ S } ( P E F = S I{ S } = L = ρ σ + λ + σ = P ( E( s U xp{ [ ρ + ( E( U λ] } i. = i= y σ xp{ s s y } y] σ s π σ s i. = i= = P ( E( s U xp{ [ ρ + ( E( U λ] } ( y + σ s s xp{ } y] σ s π σ s + σ s = P ( E( s xp{ [ ρ + ( E( U λ] } Φ = i= σ S Vi = l ( E( V r s s s whr i = + λ + ρ σ l r r E I{ S } [ ] F = P E I{ S } = r ( s s = P ( E[ P( S ] = r s s [ ( σ ] = = P E P s W s r s s P ( E[ ] = = Φ σ (. (3 s i= + σ l + ( E( U λ + r ρ + σ = = σ s σ s oghr wih ( ( a (3 w hav = = σ s s σ r = Φ ρ + λ i Φ = i= C( S P ( E [ s ( xp{ [ ( E( U ] } U ( ].. Proposiio 5 (pu-call pariy rlaioassum ha h yamics of a bo B( a a risky ass S( ar giv by ( ( mauriy a xrcis pric. h h pu-call pariy rlaio may b rwri as ρ r C( S P( S = S. Proof sic {S ( } is P marigal w hav Copyrigh Caaia Rsarch & Dvlopm Cr of Scics a Culurs 4

5 ZHENG Yigchu; YANG Yufg; ZHANG Shougag (5. Avacs i Naural Scic 8(3-5 r ( s s = V V r ( s s S = S W fi ha pu-call pariy rlaio is o affc by h jump procss of sock pric bu i is affc by coiuous ivi. W ca uspu-call pariy o fi h pric of a Europa pu opio o a sock wih h sam paramrs as arlir.wh h coiuous ivi ra ρ( = a oxplosiv couig procss {N } is Possio procss wih isiy paramr λ( = λ h rsuls of his aricl ar Mro R C s. (976 coclusio. I his papr w assum ha h ivi ha will b pai o h sharholrs urig a opio s lifim ca b pric wih craiy. W iscuss arbirag pricig wihi h opio pricig mol ur h assumpio ha h sock upo which a opio is wri pays ivi urig opio s lifim. Bcaus jumps o occur i pracic i is avaagous o cosir a mol for pric voluio ha suprimposs raom jumps o a gomric Browia moio. Assumpio ha jump procss is a cou procss ha mor gral ha Poisso procss. I is sablish ha h bhavior mol of h sock pricig procss is jump-iffusio procss. Pricig formula of Europa opio a pu-call pariy rlaio ar also giv. REFERENCES Aglo L. (9. Succssiv quaraic programmig: Dcomposiio mho. Ecyclopia of Opimizaio Ball C. & Roma A. (993. A jump iffusio mol for h Europa moary sysm. Joural of Iraioal Moy a Fiac ( Black F. & Schols M. (973. h pricig of opios a corpora liabiliis. Joural of Poliical Ecoomy 8( Cai W.. & Mao J. J. (. Polyhro o quaraic programmig a h simplx algorihm. Joural of Shaghai Elcric Powr 8( Duffi D. (996. Dyamic ass pricig hory (.. Nw Jrsay: Prico vrsiy Prss. Follmr H. & Lukr P. (. Effici hgig:cos vrsus shorfall risk. Fiac a Sochasics 4( Gill P. E. & Wog E. (. Squial Quaraic Programmig Mho. h IMA Volums i Mahmaics a Is Applicaios Harol J. & ushr. (. Jump-Diffusios wih Coroll Jumps: Exisc a Numrical Mho. Joural of Mahmaical Aalysis a Applicaios 49( Jablac M. & Poir M. (99. Opimal porfolio for a small ivsor i a mark mol wih iscoiuous prics. Appli Mahmaical Opimizaio ( ou S. S. (. Algorihm for fiig all soluios of lcpigr ss mho. Joural of iaji vrsiy 34( Mro R. (97. Opimum cosumpio a porfolio ruls i a coiuous im mol. Joural of Ecoomics hory 3( Mro R. C. (976. Opio pricig wh urlyig sock rur ar iscoiuous. Joural of Ecoomics 36( Nakao Y. (4. Miimizaio of shorfall risk i a jumpiusio mol. Saisics & Probabiliy Lrs 67( Pham H. (. Dyami L p - hgig i iscr im ur co cosrais. SIAM J.corol Opim 38( Rigr M. O. (. Opimal fiacial ivsms for ococav uiliyfucios. Ecoomics Lrs 4( Samulso P. A. & Mro R. C. (969. A compl mol of warra pricig ha maximizs uiliy. Ius.Maag. Rv YagY. F. Zhag S. G. & Xia X G. (3. Opio pricig mol wih sochasic xrcis pric. Progrss i Appli Mahmaics 5( -3. Yag Y. F. & Hao J. (3. Walh opimizaio mols wih sochasic volailiy a coiuous ivi. Iraioal Busiss a Maagm 6( Yag M. S. & Luo C.. (6. h opimizaio pricipl mho a solvig sofwar. Bijig: Scic Prss. Yua Y. X. (8. Noliar opimizaio mho. Bijig: Scic Prss. 5 Copyrigh Caaia Rsarch & Dvlopm Cr of Scics a Culurs