Black-Scholes Partial Differential Equation In The Mellin Transform Domain

Transcription

1 INTRNATIONAL JOURNAL OF SCINTIFIC & TCHNOLOGY RSARCH VOLUM 3, ISSU, Dcmbr 4 ISSN Blac-Schols Paral Dffrnal qaon In Th Mlln Transform Doman Fadgba Snday mmanl, Ognrnd Rosln Bosd Absrac: Ths ar rsns Blac-Schols aral dffrnal qaon n h Mlln ransform doman. Th Mlln ransform mhod s on of h mos olar mhods for solvng dffson qaons n many aras of scnc chnology. Ths mhod s a owrfl ool sd n h valaon of oons. W xnd h Mlln ransform mhod roosd by Pann Srvasav [7] o drv h rc of roan owr oons wh dvdnd yld. W also drv h fndamnal valaon formla nown as h Blac-Schols modl sng h convolon rory of h Mlln ransform mhod. Mahmacs Sbjc Classfcaon: 44A5, 6H3, 9G99 Kywords: Blac-Schols Modl, Blac-Schols Paral Dffrnal qaon, Dvdnd Yld, roan Oon, Mlln Transform Mhod, Oon INTRODUCTION An oon s an nsrmn whos val drvs from ha of anohr ass; hnc s calld a drvav. In ohr words an oon on an ndrlyng ass s an asymmrc conrac ha s ngoad oday wh h followng condons n h fr. Th holdr has hr h rgh, b no h oblgaon o by, as s h cas wh h roan call oon, or h ossbly o sll, as n h cas of h roan oon, an ass for a cran rc a a rscrbd da n h fr. Th Amrcan y of oon can b xrcsd a any m o ncldng h da of xry. Th dsncv fars of Amrcan oon ar s arly xrcs rvlg. Th rcng of Amrcan oons has bn h sbjc of xnsv rsarch n h las dcads. Thr s no nown closd form solon many nmrcal analyc aroxmaons hav bn roosd. Blac Schols [] blshd hr smnal wor on oon rcng n whch hy dscrbd a mahmacal fram wor for fndng h far rc of a roan oon. Thy sd a no-arbrag argmn o dscrb a aral dffrnal qaon whch govrns h volon of h oon rc wh rsc o h mary m h rc of h ndrlyng ass. Th Blac- Schols modl for rcng oons has bn ald o many dffrn commods ayoff srcrs. Fadgba Snday mmanl s crrnly a Lcrr n h Darmn of Mahmacal Scncs, Sa Unvrsy, Ado, Ngra, PH mal: mmasfad6@yahoo.com, snday.fadgba@s.d.ng Ognrnd Rosln Bosd s crrnly a Snor Lcrr n h Darmn of Mahmacal Scncs, Sa Unvrsy, Ado, Ngra, PH mal: rowbos@yahoo.com, rosln.ognrnd@s.d.ng In s of h mar crash of 987, n racc sml Blac-Schols modls ar wdly sd bcas hy ar vry asy o s [4]. W rsn an ovrvw of h Mlln ransform mhod for h valaon of oons n h conx of Blac Schols []. Th Mlln ransform s an ngral ransform namd afr h Fnnsh mahmacan Hjalmar Mlln ( ). Th Mlln ransforms n oon hory wr nrodcd by Pann Srvasav [6]. Thy drvd h xrsson for h fr bondary rc of an Amrcan ral as h lm of a fn-lvd oon. For mahmacal bacgronds, h Mlln ransform mhod n drvavs rcng som nmrcal mhods for oons valaon s [3], [5], [8], [9], [] js o mnon a fw. BLACK-SCHOLS MODL L s consdr a mar whr h rs nral ass S T, whch s govrnd by h sochasc rc, dffrnal qaon of h form ds ( ) S S dw () Whr s h dvdnd yld, r s h rslss nrs ra, s calld h volaly, T s h mary da W s calld h Wnr rocss or Brownan moon.. Drvaon of Blac-Schols Paral Dffrnal qaon Blac Schols drvd h famos Blac-Schols aral dffrnal qaon ha ms b sasfd by h rc of any drvav dndn on a non-dvdnd ayng soc. Th Blac-Schols modl can also b xndd o dal wh roan call oons on dvdnd-ayng socs. In h sql, w drv hr h Blac-Schols aral dffrnal qaon wh a dvdnd ayng soc sng orfolo aroach. W rcall from () ha; ds ( ) S d S dw whr W follows a Wnr rocss on a flrd robably, B,, F( B) n whch sac IJSTR 4

2 INTRNATIONAL JOURNAL OF SCINTIFIC & TCHNOLOGY RSARCH VOLUM 3, ISSU, Dcmbr 4 ISSN flraon F( B) S : T. Th conc of rcng roan oons on dvdnd ayng soc wll now b brfly olnd blow. Th consan connos dvdnd. In ohr words, s yld s rrsnd by ( S, ) h dvdnd aymn r n of m, whch always rrsns h sam fracon of h soc rc. Th holdr hn rcvs dvdnd aymn qal o S d whn h nrval d. As h rc of h ndrlyng ass falls by h amon of h dvdnd, h ass rc dynamcs basd on h gomrc Brownan moon modl bcoms Noc ha h las rm n (3) S d dnos h walh addd o h orfolo d o h dvdnd ylds by alyng h no-arbrag argmn, h hdgd orfolo shold arn h rs-fr nrs ra, so ha d rd (7) whr f f S S (8) ds S ( ) d dw For vry ass hld () Sd s rcvd. Th holdr of h orfolo, who holds asss, arns an amon qal o dvdnd aymn qals o S d n h nrval d. Th chang n val of h orfolo s gvn by d df ds S d (3) Sbs (6) (8) no (7) w hav f s f f s d s s f r f S d S qaon (9) bcoms (9) Whr f f s f f df S d S dw S S S f S (4) f f S f r S S S rf () qaon () s calld Blac-Schols aral dffrnal qaon on a dvdnd ayng soc. f ( S, ) f s h far rc of h ndrlyng ass. Sbsng () (4) no (5), w hav f f S f d S d S S f f S dw Sd Sdw S S f S d S Thrfor, f S f f d S d (5) S S (6) 3 TH MLLIN TRANSFORM MTHOD FOR TH VALUATION OF UROPAN PUT OPTION ON A DIVIDND PAYING STOCK Sng f ( S, ) oon of h form n qaon (), w hav roan ( S, ) ( S, ) r S S P( S, ) S S r( S, ) wh bondary condons () IJSTR 4

3 INTRNATIONAL JOURNAL OF SCINTIFIC & TCHNOLOGY RSARCH VOLUM 3, ISSU, Dcmbr 4 ISSN lm ( S, T) on [, T) S (), max K S, P S T J S K S on [, ) (3) ( S, ) ( S, ) M M r S s S ( S, ) M s (9) M r ( S, ) M () r T (, ) K on [, ) Th Mlln Transform for roan oon, by (, ) (, ) w w S S ds T (4) s gvn (5) whr w s a comlx varabl wh R( w). Th nvrson of h Mlln ransform s also gvn by c w ( S, ) ( w, ) S dw c (6) qaon (6) holds vrywhr on (, ), whr ( S, ) ( w, ) ar calld a Mlln ransform ar wh h followng condons ( S, ) O(), for ( S, ) O( S ), for S (7) S (8) Tang h Mlln ransform of (), w hav ( S, ) ( S, ) r S S M () M S ( S, ) r (, ) S S From h rors of h Mlln ransform w hav ha,, S w M S, M r S wr w, S S S, M w w w, S M r ( S, ) r w,, M () Sbsng () no (9) ylds, () w wr, w r ww w, w, Sng ( w, ) Thrfor, w r w w r IJSTR 4

4 INTRNATIONAL JOURNAL OF SCINTIFIC & TCHNOLOGY RSARCH VOLUM 3, ISSU, Dcmbr 4 ISSN w r w w r ln ( w, ) Gw w, r r w w Sng r z z r (3) () () Thrfor, w, ( w,) Whr, G( w) (7) w s a consan ha dnds on h bondary condons. Now, l s consdr h rmnal condon gvn by (3) as, max, S J S K S K S So, P G( w) T ( w, o) J ( w, ) J w, (8) Whr Sbsng () (3) no (), hn w hav w w( z ) z Sng ( ) ( ) (4) G w w w z z (5) Sbsng (5) no (4) ylds G( w) (6) Sarang varabls n (6) on ngraon, w hav w, G w (,) w w K ww ( ) (9), J w J w qaon (9) s calld h Mlln ransform of (3), hn (7) bcoms G( w)( T ) (3),, w J w Usng h nvrs of h Mlln ransform, w can wr for h rc of a roan oon as c w P S, w, S dw (3) c Sbs qaon (3) no (3) w hav c GwT w P( S, ) J w, s dw c (3) IJSTR 4 3

5 INTRNATIONAL JOURNAL OF SCINTIFIC & TCHNOLOGY RSARCH VOLUM 3, ISSU, Dcmbr 4 ISSN whr c, a consan, w R( w) S,,, T 3. Solon of Blac-Schols Paral Dffrnal qaon wh a Dvdnd Payng Soc n h Mlln Transform Doman W drv hr h solon of Blac-Schols aral dffrnal qaon sng Mlln ransform by mans of algbrac ransformaons L W hav ha T w z (33) T G w T w w z z z z T w z T w z Thrfor, T Gw w z (34) Ths, w can wr for h rc by sbsng (34) no (3), w hav z c w w S, J w, s dw c Now, * w s h Mlln ransform of w w w (35) (36) S S ds ln S 4J w, R( ) (37) Jw S ds J J W g S S ( S, ) T ln S T (38) From h convolon rory of h Mlln ransform, w can wr ha S S, J d whr J( ) ( K ) S S T S ln T (39) (4) (4) Sbsng (4) (4) no (39), hn h roan owr rc can hrfor b xrssd as z S ln T ( ) S S, K d T (4) qaon (4) bcoms Usng h ransformaon n [], w hav IJSTR 4 4

6 INTRNATIONAL JOURNAL OF SCINTIFIC & TCHNOLOGY RSARCH VOLUM 3, ISSU, Dcmbr 4 ISSN S ln z T K S S, K d T S ln z T K S d T (43) z T K * S d T (49) * * S, S ln valang, w hav rscvly * * (5) whr T, qaon (43) can b xrssd as z r, z s z ln K S, KS d T U T S ln z T K S d T S, KS S T (44) (45) whr dno frs scond ngrals of (44) rscvly. To vala, w s h followng varabls blow S T (46) ln T S ln ( ) T T (47) z T K S K d T (48) S ln K * r T N d * T S N d Sbsng qaons (5) (5) no (5), w hav rt T, S K N d S N d whr S ln r ( T ) d T S ln r T d d T T (5) (5) (53) (54) qaon (53) s calld Blac-Schols modl for h solon of roan oon sng h Mlln ransform mhod. Th abov rsl can b smmarzd n h horm blow as follows: Thorm : L S b h rc of h ndrlyng ass, K b h sr rc, r b h rs nrs ra T b h m o mary. Usng h convolon rory of h Mlln ransform gvn by S S, J d W hav h Blac-Schols formla for h valaon of roan owr oon as rt T, S K N d S N d whr h aramrs d d ar gvn by IJSTR 4 5

7 INTRNATIONAL JOURNAL OF SCINTIFIC & TCHNOLOGY RSARCH VOLUM 3, ISSU, Dcmbr 4 ISSN S ln r ( T ) d T S ln r T d d T T Th normal dsrbon s gvn by N( x) d x [8] Vaslva O. (9), A Nw Mhod of Prcng Ml-Oons sng Mlln Transforms Ingral qaons, Masr's Thss n Fnancal Mahmacs, School of Informaon Scnc, Comr lcrcal ngnrng, Halmsa Unvrsy. [9] Yabovch S.B. Ngyn T.H. (99), Th Dobl Mlln-Barns Ty Ingrals hr Alcaons o Convolon Thory, Srs on Sov Mahmacs, World Scnfc, 99. [] Znb A.. Roah R.A. (), Analycal Solon for an Arhmc Asan Oon sng Mlln Transforms, Vol. 5, CONCLUSION In hs ar, w hav consdrd h Blac-Schols aral dffrnal qaon n h Mlln ransform doman. W hav sablshd a formla for h valaon of roan owr oon sng h Mlln ransform mhod on a dvdnd yld conssng of sngl ngral. W also show how o drv h Blac-Schols valaon formla for roan owr oon by mans of h convolon rory of h Mlln ransform mhod. Rfrncs [] Blac F. Schols M. (973), Th Prcng of Oons Corora Labls. Jornal of Polcal conomy, Vol. 8, No. 3, [] rdly A, al. (954), Tabls of Ingral Transforms, Vol. -, Frs don, McGraw-Hll, Nw Yor. [3] Fadgba S.. C. R. Nwozo (4), On h Comarav Sdy of Som Nmrcal Mhods for Vanlla Oon Valaon, Commncaon n Ald Scncs, Vol., No.. [4] Ingbr L. Wlson J.K. (), Sascs Mchancs of Fnancal Mars: xrmnal Modfcaons o Blac-Schols, Mahmacal Comr Modllng, Vol. 3, No. 8-9, [5] Nwozo C.R. Fadgba S.. (), Mon Carlo Mhod for Prcng Som Pah Dndn Oons, Inrnaonal Jornal of Ald Mahmacs, Vol. 5, No. 6, [6] Pann R. Srvasav R.P. (4), Prcng Pral Oons sng Mlln Transforms, Ald Mahmacs Lrs, Vol. 8, , do:.6/j.aml [7] Pann R. Srvasav R.P. (4), Oon Prcng wh Mlln Transforms, Mahmacal Comr Modllng, Vol. 4, 43-56, do:.6/j.mcm IJSTR 4 6