A New Approach for Solving Boundary Value Problem in Partial Differential Equation Arising in Financial Market

Transcription

1 Applied Mahemaics, 6, 7, Published Online May 6 in cires. hp://.scirp.org/journal/am hp://dx.doi.org/.436/am A Ne Approach for olving Boundary Value Problem in Parial Differenial Equaion Arising in Financial Marke Fadugba unday Emmanuel *, Emeka Helen Oluyemisi Deparmen of Mahemaical ciences, Ekii ae Universiy, Ado Ekii, Nigeria Deparmen of Mahemaical and Physical ciences, Afe Babalola Universiy, Ado Ekii, Nigeria Received April 6; acceped 3 May 6; published 6 May 6 Copyrigh 6 by auhors and cienific Research Publishing Inc. This ork is licensed under he Creaive Commons Aribuion Inernaional License (CC BY). hp://creaivecommons.org/licenses/by/4./ Absrac In his paper, e presen a ne approach for solving boundary value problem in parial differenial equaion arising in financial marke by means of he Laplace ransform. The resul shos ha he Laplace ransform for he price of he European call opion hich pays dividend yield reduces o he Black-choles-Meron model. eyords Black-choles-Meron Model, Boundary Value Problem, European Call Opion, Financial Marke, Laplace Transform. Inroducion An opion is a conrac ha gives he righ (no an obligaion) o is holder o buy or sell some amoun of he underlying asse a a fuure dae for a prescribed price. The underlying asses include socks, sock indices, deb insrumens, commodiies and foreign currency. A call opion gives is holder he righ o buy he underlying asse, hereas a pu opion gives is holder he righ o sell. Vanilla opions are acively raded on organized exchanges. They are also subjec o cerain regulariy and sandardizaion condiions. Vanilla opions can be classified as American opions and European opions. An American opion gives a financial agen he righ, bu no obligaion o buy or o sell he underlying asses on or prior o he expiry dae a he specified price called he exercise price. European opion is an opion ha can be exercised only a he expiry dae ih linear payoff. European opion comes in o forms namely European call and pu opions. * Corresponding auhor. Ho o cie his paper: Emmanuel, F.. and Oluyemisi, E.H. (6) A Ne Approach for olving Boundary Value Problem in Parial Differenial Equaion Arising in Financial Marke. Applied Mahemaics, 7, hp://dx.doi.org/.436/am

2 A European call opion is an opion ha can be exercised only a expiry and has a linear payoff given by he difference beeen underlying asse price a mauriy and he exercise price. The payoff for he European call opion is given by ( T, ) ( T ) C T +. () A European pu opion is an opion ha can be exercised only a expiry and has a linear payoff given by he difference beeen he exercise price and underlying asse price a mauriy. The payoff for he European pu opion is given by ( T, ) ( T ) P T +. () The revoluion on derivaive securiies boh in exchange markes and in academic communiies began in he early 97 s. Ho o raionally price an opion as no clear unil 973, hen Black and choles published heir seminal ork on opion pricing in hich hey described a mahemaical frame ork for finding he fair price of a European opion (see []). Moreover, in he same year, [] exended he Black-choles model in several imporan ays. ince is invenion, he Black-choles formula has been idely used by raders o deermine he price of an opion. Hoever his famous formula has been quesioned afer he 987 crash. One of he main concerns abou financial opions is ha he exac values of opions are. For he simples model in he case of consan coefficiens, an exac pricing formula as derived by Black and choles, knon as he Black-choles formula. Hoever, in he general case of ime and space dependen coefficiens he exac pricing formula is no ye esablished, and hus numerical soluions have been used (see [3]). There are many exhausive exs and lieraures in his subjec area such as [4]-[], jus o menion a fe. In his paper, e presen a ne approach for solving boundary value problem in parial differenial equaion arising in financial marke via he Laplace ransform. The res of he paper is organized as follos: ecion presens he Black-choles-Meron parial differenial equaion for he price of European call opion hich pays a dividend yield. In ecion 3, e consider he Laplace ransform and some of is fundamenal properies. ecion 4 presens he Laplace ransform for solving boundary value problem in parial differenial equaion arising in financial marke. We also sho ha our resul reduces o Black-choles-Meron like formula. ecion 5 concludes he paper.. The Black-choles-Meron Parial Differenial Equaion We consider a marke here he underlying asse price, T is governed by he sochasic differenial equaion of he form d r d d + d W, < < (3) here is he volailiy, r is he riskless ineres rae, d is he dividend yield and W is a one-dimensional Wiener process. andard arbirage argumens sho ha any derivaive u(, ) rien on mus saisfy he parial differenial equaion of he form eing u(, ) C(, ) u, u, u, + ( r d ) + ru (, ). (4) in (4), hen e have he Black-choles-Meron parial differenial equaion for he price of European call opion given by ih boundary condiions C, C, C, + ( r d ) + rc (, ) (, ) as on [, ) (, ) as on [, ) C T (6) C T (7) (5) 84

3 and final ime condiion given by + ( T, ) ( T ) ( T ) on [, ) C T f. (8) Equaion (7) saes ha he opion is orhless hen he sock price is zero. 3. The Laplace Transform and Is Fundamenal Properies Le h( x ) be a piece-ise coninuous funcion on every closed inerval { x [ ab, ] x [, )} h: { x [, ) }, h: x h( x) such ha and. Then L ( h( x) ) H( ),for he Laplace ransform of h( x ) and is defined as ( ) e x d,, > here exiss is called L h x H h x x (9) henever he inegral exiss. Conversely, he inverse Laplace ransform of H( ) is defined as L ( H( ) ) h( x) H( ) c > πi Le h( x ) be a piece-ise coninuous ih he Laplace ransform he Laplace ransform hold. ) Lineariy of he Laplace Transform c+ i x e d,. () c i ( ) + ( ) + ( + ) L h x. The fundamenal properies of L e x ah x L bg x ah bg ah x bg x dx. () Equaion () is inermediae from he definiion and he lineariy of he definie inegral. ) caling Propery x L ( ax) h( ax) e d x H, a > a a () 3) hifing Propery 4) Commuaiviy Propery The Laplace ransform is commuaive. i.e. ( e ax ) e ax e x d ( ) L x H a (3) ( ) ( ) ( ) L g x * L h x G * H g x h d 5) The Laplace Transforms on Differeniaion x x d * ( )* ( ) h x g H G L h x L g x Le h( x),for x > be a differeniable funcion ih he derivaive h x M x X hen dh x dx (4) being coninuous. uppose ha α x here exis consan M and X such ha e,, b dh( x) x dh( x) x L ( h( x) ) H ( ) e dx lim e dx H ( ) h( ). (5) Noe ha he condiion 6) Convoluion Propery dx b α x dx b h x Me, x X lim h b e, for > α. (6) b 84

4 Theorem : Convoluion Theorem and Le L( g( x) ) L( h( x )) denoe he Laplace ransforms of g( x) and h( x ), respecively. Then he produc given by L( f ( x) ) L( g( x) h( x) ) is he Laplace ransform of he convoluion of g( x) and h( x ) is denoed by f ( x) ( g* h)( x) and he inegral represenaion x f x g* h x g x h x s ds f ( x) ( h* g)( x) h( x) g( x s) ds x We presen some of he resuls on he exisence and uniqueness of he Laplace ransform belo. Theorem : Exisence of Laplace Transform Le h( x ) be a pieceise coninuous funcion on [, R ] (for every R > ) and have an exponenial order h x M L h x is defined for > β, i.e. a infiniy ih e ax. Then, he Laplace ransform ( ) { } { β} Domain L ( h( x) ) >. Theorem 3: Uniqueness of Laplace Transform Le g( x) and ions ih an exponenial order a infiniy. Assume ha L g( x) L h x, hen g( x) h( x) x [ P] for every P >, excep may be for a finie se of poins. Relaion o he Mellin and he Fourier Transformaions (7) h x be o pieceise coninuous func- ( ),,, Laplace ransformaion is closely relaed o an exended form of oher popular ransforms, paricularly Mellin and Fourier. Boh can be obained hrough a change of variables. By seing z z x e,dx e dz. (8) The Laplace ransform (9) yields M h z e, F h z e, i L h x, (9) here (.), (.) and (.) ( ) ( ) ( ) M F L denoe he Mellin ransform, he Fourier ransform and he Laplace ransform respecively 4. Laplace Transform for olving Boundary Value Problems in Parial Differenial Equaion Arising in Financial Marke By change of variables T, (5) becomes C(, ) C(, ) C(, ) + ( r d ) + rc (, ) () ih boundary condiions C, as on, T () and final ime condiion given by ( ) [ ) (, ) as on [, ) C T () + (, ) ( ) on [, ) C. (3) Le he Laplace ransform for he price of he European call opion be defined as L C, f (4) ( ( )) ( ) and he inverse Laplace ransform for he price of he European call opion be given by ( ( )) (, ) L f C (5) 843

5 here f is he Laplace ransform ih parameer. Taking he Laplace ransform of (5) using (4) e have ha here L is he Laplace operaor and L C, + ( r d ) C, + C, L rc ( (, ) ) (, ) C L f C f + ( ( ) (, )) ( ) ( ) (, ) C f L ( r q) ( r d) ubsiuing (7), (8), (9) and (3) ino (6) yields (, ) C f L ( (, )) ( ) + (6) (7) (8) (9) L rc rf. (3) + f f f r d rf implifying furher and rearranging erms in (3) e have ha f f + r d r + f We consider he folloing o cases as follos. CAE I For, (3) becomes f f + r d r + f The general soluion o (33) can be obained as here f ( ) and f ( ) form c p c p +. (3). (3). (33) f f + f (34) are he complemenary soluion o he homogeneous par of (33) hich is of he and he paricular soluion respecively. We assume ha he soluion o (33) is of he form and f f + + ( r d) r f The firs and he second derivaives of (36) are obained as f (35) f B α. (36) f ( ) ( ) α B α α α B α ubsiuing (36), (37) and (38) ino (35), and simplifying furher, e have ha (37). (38) 844

6 olving (39), e obain he folloing roos α + α ( + ). (39) α B r d r rd + rd + r+ α rd rd + r+ α Hence he complemenary soluion o he homogeneous par of (33) is obained as α α here α and α are given by (4). For he paricular soluion of (33), e assume ha (4) fc B + B, α α (4) f f f j + l j and. (4) p p p Using (4) and (33), and equaing he coefficiens of erms, e obain j l ( + d) ( r + ) ubsiuing (43) and (44) ino he paricular soluion, e have ha ubsiuing (4) and (45) ino (34) f p ( ) ( + d) ( r + ) α α (43). (44) ( + d) ( r + ). (45) f ( ) B + B +, α α. (46) Equaion (46) is he general soluion o (33) for. CAE II For, (33) becomes ih f f + +. ( r d) r f Folloing he above procedures, he general soluion o he las equaion is obained as α α f B + B, α α (47) rd + rd + r+ α rd rd + r+ α 845

7 Equaion (47) coincides ih he complemenary soluion of (33) given by (4). In he case of, e se B o ensure he boundedness of he derivaive f ( ). In he case of, B o ensure ha he opion s value approaches zero as he sock price goes o zero. The soluions for he o cases equaions (46) and (47) become α f( ) B +, α α ( + d) ( r + ) (48) f B α, α α. (49) We an he opion pricing funcion o be coninuous and differeniable a he ransiion poin, Therefore, he values of he funcion and heir firs derivaives from (48) and (49) mus equal o each oher. These condiions can be used o solve for B and B. The funcion values and derivaives a from (48) and (49) are given by α f( ) B + (5) d + r + and and ( ) α B α + f d + (5) f, B α (5) f ( ) Bα α eing (5) (5) and (5) (53) and solving furher, e obain B ( α ) ( α α ) α ( r + ) d + α. (53) ( α ) ( α α ) α B ( r + ) d + α ubsiuing (54) and (55) ino (49) and (48), e have ( α ) α α ( α α ) (54). (55) α f( ) +, α α ( r + ) d + + d r + ( α ) α ( α α ) α α f( ), α α ( r + ) d + respecively. Equaions (56) and (57) can also be rien as f ( ) ( α ) α α ( α α) ( α ) α, α α, for α ( α α ) α +, α α, for ( r + ) d + + d r + α ( r + ) d + Equaion (58) is he Laplace ransform of he price of European call opion hich pays a dividend yield. Theorem 4 (56) (57) (58) 846

8 If, α α, hen he Laplace ransform of he price of European call opion ih dividend yield given by ( α ) α ( α α ) α f ( ) f ( ) reduces o he Black-choles-Meron valuaion formula α ( r + ) d + BM (, ) e d r e (58a) C N d N d (58b) ih ln r d + + d ln r d + d d by means of he Laplace ransform of he form here k g+ a g m+ b ak e L e N e c g + a a+ g + a (58c) r d + r d + ( r + ) α, r d r d + ( r + ) α, b m a and k and bcm,, and g are arbirary consans. c c Proof: From (57) and (58a) e can rie ha eing Therefore, (59) becomes Le us firs consider he erm ( α ) α ( ) α f f α α α ( r + ) ( α α) d + α α A A α α ( r + ) ( α α ) ( α ) α ( d + ) ( α α ) α α (58d). (59) (6). (6) f f A + A. (6) α α A. (63) r r α α ( + ) ( α α) + α α α 847

9 Using he values of and Therefore (63) yields r d r d + r + α ( α α ) r d + ( r + ). α r d + r d + ( r + ) A ( r ) + r d + ( r + ) Thus, r d r d ( r ) ln α e ( r + ) r d + r + ln α e r d + ( r + q) r d + r d + r + ln α e r d + r d + ( r + ) A ( r + ) r d + ( r + ) r d + r d + ( r + ) ubsiuing he value of α ino (64) and simplifying furher, e obain ln rd + ( r+ ) r d ln e e A r d r d r d ( r + + ) + + ( r + ) (64) (65) 848

10 Comparing (65) ih (58c), e have ha,, ln, r d b g ra,, c ln m k. c b r d c m Taking he inverse Laplace ransform of (65), e obain here We also consider he erm r ( ) e r (66) A e N d N d (67) A d ln r d +. (68) ( α ) α ( d + ) ( α α ) ( α ) ( α α ) α d + ubsiuing he values of α and α ino (69) yields A implifying (7) furher, e obain A α r d ( r ) r d ln ln. (69) e e r d r d + r d ( r ) ( r ) ln r d r d ln (7) e e. (7) r d + r d + r d Once again e compare (7) ih (58c), e deduce ha 849

11 +,, ln, r d b + g da,, c ln m k c b r d c m Taking he Laplace ransform of (7), e have ha e d (7) A N d (73) here ln r d + + d. (74) The inverse Laplace ransform of (6) is obained as (, ) (, ) ubsiuing (67), (68), (73) and (74) ino (75) yields C C A + A. (75) d r (, ) (, ) e e (, ) C C N d N d C BM ln r d + + d ln r d + d d This complees he proof. Theorem 5 If, α α, hen he Laplace ransform of he price of European call opion ih dividend yield given by (, ) C ( α ) α α ( α α ) α ( r + ) d + + d r + + reduces o he Black-choles-Meron valuaion formula given by BM d r (, ) e e (76) C N d N d (76a) ih ln r d + + d ln r d + d d by means of he Laplace ransform given by 85

12 here k g+ a g m+ b ak e L e N e c g + a a+ g + a (76b) r d + r d + ( r + ) α, r d r d + ( r + ) α, b m a and k and bcm,, and g are arbirary consans. c c Remark ) The proof of Theorem 5 follos from Theorem 4, since (56) and (57) have he same inverse Laplace ransforms. The above resuls sho ha he prices of European call opion ih dividend yield given by (56) and (57) coincide ih he Black-choles-Meron model given by (58b) by means of (58c). 5. Conclusion Finance is one of he fases developing areas in he modern corporae and banking orld. In his paper, e have considered he boundary value problem in parial differenial equaion arising in financial marke. We used a ne approach for solving he Black-choles-Meron parial differenial equaion for he price of European call opion hich pays a dividend yield by means of he Laplace ransform. The same approach can be used for European pu opion ih dividend paying sock. The resuls sho ha he Laplace ransform for he price of he European call opion ih dividend paying sock coincides ih he Black-choles-Meron model; i is very effecive and is a good ool for solving parial differenial equaions arising in financial marke and oher areas such as engineering and applied sciences. References [] Black, F. and choles, M. (973) The Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy, 8, hp://dx.doi.org/.86/66 [] Meron, R.C. (973) Theory of Raional Opion Pricing. Bell Journal of Economics and Managemen cience, 4, hp://dx.doi.org/.37/3343 [3] Lee, H. and heen, D. (9) Laplace Transformaion Mehod for he Black-choles Equaion. Inernaional Journal of Numerical Analysis and Modelling, 6, [4] Cohen, A.M. (7) Numerical Mehods for Laplace Transform Inversion. pringer, Ne York. [5] Nozo, C.R. and Fadugba,.E. (4) Performance Measure of Laplace Transforms for Pricing Pah Dependen Opions. Inernaional Journal of Pure and Applied Mahemaics, 94, hp://dx.doi.org/.73/ijpam.v94i.5 [6] Fadugba,.E. and Ogunrinde, R.B. (4) Black-choles Parial Differenial Equaion in he Mellin Transform Domain. Inernaional Journal of cienific and Technology Research, 3, -6. [7] Lee, R. (3) Opion Pricing by Transform Mehods: Exension, Unificaion and Error Conrol. anford Universiy, Working Paper. [8] Perella, G. and uo,.g. (4) Numerical Pricing of Discree Barrier and Lookback Opions via Laplace Transforms. Journal of Compuaional Finance, 8, -37. [9] Zieneb, E. and Rokiah, R.A. (3) olving an Asian Opions PDE via Laplace Transform. cience Asia, 39, hp://dx.doi.org/.36/scienceasia [] himko, D.C. (99) Finance in Coninuous Time. A Primer, olb Publishing. (76c) 85