Numerical Evaluation of European Option on a Non Dividend Paying Stock

Transcription

1 Inernaonal Journal of Compuaonal cence and Mahemacs. IN olume Number 3 (00) pp Inernaonal Research Publcaon House hp:// Numercal Evaluaon of European Opon on a Non Dvdend Payng ock bha Mshra and K.R. Pardasan Deparmen of Mahemacs M.A.N.I.T Bhopal (M.P.) Inda E-mal: vbha_67@redffmal.com kamalrp@homal.com Absrac In hs paper an aemp has been made o numercally evaluae he value of European opon employng Black-scholes formula. The mporan parameers lke me volaly srke prce neres rae and spo prce have been ncorporaed n he model. Approprae boundary and nal condons have been framed usng fnancal condons of he sock marke. The numercal mehods especally he mplc and explc schemes have been employed o solve Black-choles PDE. The resuls obaned are used o sudy he relaonshps among varous parameers. Keywords: European opon non-dvdend srke prce volaly. Inroducon Black and choles [9] proposed an explc formula for evaluang European call opons whou dvdends whch s sll exensvely used wh underlyng. The numercal soluon of hs equaon has been of paramoun neres due o he governng paral dfferenal equaon whch s very dffcul o generae sable and accurae soluons [94]. A European call opon gves s holder he rgh (bu no he oblgaon) o purchase from he wrer a prescrbed asse for a prescrbed prce a a prescrbed me n he fuure. The prescrbed purchase prce s known as he exercse prce or srke prce (E) and he prescrbed me n he fuure s known as he expry dae (T). The drec oppose of a European call opon s a European pu opon [3 4 5 ]. A European pu opon gves s holder he rgh (bu no he oblgaon) o sell o he wrer a prescrbed asse for a prescrbed prce a a prescrbed me n he fuure. Therefore he value of he opon a expry known as he pay-off funcon s ( T) (-E). Numercal mehod of long sandng for general parabolc free boundary problem s he me dscree. If dfferenal equaon s used o model he value of he opon

2 6 bha Mshra and K.R. Pardasan hen he resulng problem no longer s formulaed for a consan coeffcen scalar (hea) equaon bu for a one dmensonal non lnear parabolc equaon[ 0]. Fne dfference heory has a long hsory and has been appled for more han 00 years o approxmae he soluons of paral dfferenal equaons n he physcal scences and engneerng. Wha s he relaonshp beween FDM and fnancal mahemacs? To answer hs queson we noe ha he behavor of a sock (or some oher underlyng) can be descrbed by a sochasc dfferenal equaon [6 7]. Then a conngen clam ha depends on he underlyng s modeled by a paral dfferenal equaon n combnaon wh some nal and boundary condons. olvng hs problem means ha we have found he value for he conngen clam [3]. Problem In hs paper we consder how mgh be used o value an European opon on a nondvdend payng sock. The Black-choles paral dfferenal equaon for European opon on non-dvdend payng sock s. r σ r () uppose ha he lfe of he opon s T. we dvde hs no N equally spaced T nervals of lengh. A oal of N mes are consdered..e. N T.mlarly max s a sock prce of he opon. We max defne and consder a oal of M equally spaced sock prce of he M opons max. The ( ) pons on he grd s he pon ha correspondng o and sock prce. We wll use he varable o denoe he value of he opon a he ( ) pon. The followng wo mehods are employed for solvng he fundamenal PDE numercally. The mplc Fne Dfference mehod (The backward Euler mehod) The explc Fne Dfference mehod (The Euler mehod) Implc Fne Dfference Mehod: - For he neror pon ( ) on he grd can be approxmaed as () OR (3) Equaon () s known as he forward dfference approxmaon and equaon (3) s known as he backward dfference approxmaon. We use more symmercal

3 Numercal Evaluaon of European Opon 63 approxmaon by averagng he wo. (4) For we wll use a forward dfference approxmaon so ha he value a me s relaed o he value a me (). (5) The backward dfference approxmaon for a he ( ) pon s gven by equaon () he backward dfference a he () pon s Hence a fne dfference approxmaon for a he ( ) pon s (6) ubsung equaons (4) (5) and (6) no he dfferenal equaon () and gves r r σ For 3.. M- and 03 N-rearrangng erms we oban c b a (7) Where r a σ r b σ r c σ A program has been developed n MATLAB 7.5 for hs mplc fne dfference scheme o evaluae he fnancal opons.

4 64 bha Mshra and K.R. Pardasan Explc mehod Here we use a backward dfference approxmaon of he me dervave. Ths s he dea behnd he explc mehod whch approxmaes If hs approxmaon s appled n conuncon wh (4) and (6) n he PDE () hen r σ r For 3.. M- and 03 N- and rearrangng erm we oban Here a b c! (8) r σ r a r b ( σ ) r σ r c The program has been developed for explc fne dfference mehod n Malab7.5 o evaluae he fnancal opons. Resuls and Dscusson The value of varous parameers lke me volaly srke prce sock prce and neres rae whch have been used for numercal compuaons are gven n able (). The numercal resuls have been obaned for European call and pu opons by he wo numercal mehods namely Implc and Explc fne dfference mehod. The fgure () & () represens he resuls obaned by mplc mehod whle fgure 3 & 4 represens he resuls obaned by explc mehod. In Fg and 3 we observe ha when me s. yrs (73days) opon prce sar ncreasng when sock prce s 50 and opon prce goes an ncreasng when me s yr (365 days). we see ha hs relaon among opon prce sock prce and me s lnear. Furher he relaonshp beween opon prce and sock prce for pu opon n fgure () and (4) are us oppose of he same relaonshp for call opon n fgure () and (3). There s no sgnfcan dfference n he resuls obaned by he mplc and explc mehod. uch models can be developed o sudy relaonshp among he fnancal parameers for predcon of sock markes.

5 Numercal Evaluaon of European Opon 65 Table : alues of parameers for European opons [6789]. Parameers ymbol Numercal value ock prce 00 rke /Exercse Prce E 50 Ineres rae R 0% Opon Maury T monhs yr olaly σ.06 Fgure : Graph among Opon Prce sock Prce and me for call Opon by Implc mehod. Fgure : Graph among Opon Prce sock Prce and me for pu opon by mplc mehod. Fgure 3: Graph among Opon Prce sock Prce and me for call opon by Explc mehod. Fgure 4: Graph among Opon Prce sock Prce and me for pu opon by explc mehod. References [] Jula Ankudnova & Mahas Ehrhard(008) On he numercal soluon of nonlnear Black choles equaons Compuers and Mahemacs wh Applcaons vol

6 66 bha Mshra and K.R. Pardasan [] Hull John C. Opons (006) Fuures and Oher Dervaves 6h ed. Upper addle Rver NJ: Prence-Hall chaps. 9. [3] Angelos Dassos & Jayalaxshm Nagaradasarma 004 Prcng Of Asan Opons on Ineres Raes n he CIR Model Proceedng (437) Fnancal Engneerng and Applcaons. [4] Andrew M. Chsholm Dervaves Demysfed (004). A ep-by-ep Gude o Forwards Fuures waps and Opons John Wley & ons Ld. [5] Desmond J. Hgham 004 An Inroducon To Fnancal Opon aluaon Mahemacs ochasc And Compuaon Cambrdge unversy Press. [6] Jaksa Cvanc and Fernando Zapaero (004) Inroducon To The Economcs And Mahemacs Of Fnancal Marke The MIT PRE Cambrdge. [7] Karhan Akcoglu k Mng-Yang Kao & huba Raghavan feb (004) Fas Prcng of European Asan Opons wh Provable Accuracy: ngle-sock and Baske Opons. [8] Leser Ingber & Jennffer K. Wlson (004) ascal Mechancs of Fnancal Markes: Exponenal Modfcaons o Black-choles. [9] N. ukhomln (004) ymmery and new soluons of he Black choles equaon Bolen de la Asocacon Maemaca enezolana () [0] Cho. and Marcozz M.D. (00) A Numercal Approach o Amercan Curreny Opon aluaon Journal of Dervaves vol. 9. [] M. Yor (00) Exponenal funconals of Brownan moon and relaed processes prnger Fnance. prnger-erlag Berln. [] G. Alobad (000) Amercan opons and her sraeges Ph.D. hess Unversy of Wesern Onaro London Canada. [3] Rober Kolb John Wley & sons (995) Inc. New York Chcheser undersandng opons [4]. Heson (993) A closed-form soluon for opons wh sochasc volaly wh applcaons o bond and currency opons The Revew of Fnancal udes 6: [5] J.C. Cox.A. Ross M. Rubnsen 7 (979) Opon prcng:a smplfed approach Journal of Fnancal Economcs [6] hp:// [7] hp:// [8] hp:// [9] hp:// Hoadley Tradng & Invesmen Tool