A New Technique for Solving Black-Scholes Equation for Vanilla Put Options

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1 Briish Journal of Mahemaics & Compuer Science 9(6): , 15, Aricle no.bjmcs ISSN: SCIENCEDOMAIN inernaional A New Technique for Solving Blac-Scholes Equaion for Vanilla Pu Opions S. E. Fadugba 1*, R. B. Ogunrinde 1, S. C. Zelibe and C. Achudume 3 1 Deparmen of Mahemaical Sciences, Eii Sae Universiy, Ado Eii, Nigeria. Deparmen of Mahemaics, Federal Universiy of Peroleum Resources, Effurun, Nigeria. 3 Deparmen of Mahemaics, Universiy of Ibadan, Oyo Sae, Nigeria. Aricle Informaion DOI: /BJMCS/15/18134 Edior(s): (1) Dragoş - Păru Covei, Deparmen of Mahemaics, Universiy Consanin Brâncuşi of Târgu-Jiu, România. Reviewers: (1) Brigh O. Osu, Deparmen of Mahemaics, Abia Sae Universiy, Nigeria. () Anonymous, Mexico. Complee Peer review Hisory: hp://sciencedomain.org/review-hisory/977 Original Research Aricle Received: 6 April 15 Acceped: 18 May 15 Published: 15 June 15 Absrac In his paper we presen a new echnique for solving Blac-Scholes equaion for vanilla pu opions using he Mellin ransform mehod. eywords: Blac-Scholes equaion; Mellin ransform mehod; vanilla pu opion. 1 Mahemaics Subjec Classificaion: 44A15, 91G, 91G3. 1 Inroducion Vanilla opion is a conrac ha gives he holder he righ, bu no he obligaion, o buy or sell an insrumen a a specified srie price on or before a specified dae. The common examples of vanilla opions are American and European opions. American opions are opions ha allow opion holders o exercise he opion a any ime prior o and including is mauriy dae, hus increasing he value of he opion o he holder relaive o European opions, which can only be exercised a mauriy. The majoriy of exchange-raded opions are American [1]. Nowadays, he Blac-Scholes equaion derived by [] is widely used in he field of mahemaical finance. Despie he success of he Blac-Scholes model on hedging and pricing coningen claims, [3] noed early ha opions quoed on he mares differ sysemaically from heir prediced values, which led up o quesioning he disribuional assumpions based on geomeric wiener process [4]. Alernaive approach for *Corresponding auhor: emmasfad6@yahoo.com;

2 Fadugba e al.; BJMCS, 9(6): , 15; Aricle no.bjmcs he soluion of he Blac-Scholes parial differenial equaion for European call opion which pays dividend yield was considered by [4]. [5] inroduced he Mellin ransforms in he heory of opion pricing. They derived he expression for he free boundary and price of an American perpeual pu as he limi of finie lived opions. In his paper we shall derive a new echnique which does no require ransformaion of variables for solving Blac-Scholes equaion for vanilla pu opions using he Mellin ransforms. See ([6-17]) for he mahemaical bacgrounds of he Mellin ransform mehod for opions valuaion and oher approaches for he soluion of Blac-Scholes equaion. The res of he paper is srucured as follows. Secion inroduces he Mellin ransforms. In secion 3 we presen a new echnique for solving Blac-Scholes equaion for vanilla pu opions such as European pu opions. Secion 4 concludes he paper. The Mellin Transforms The Mellin ransform denoed by M is a complex valued funcion defined on a verical srip in he v -plane and is given by M v1 ( f ( y), v) F( v) f ( y) y dy (1) where f ( y) is a funcion defined on he posiive real axis y (, ). The funcion F (v) is called he Mellin ransform of f ( y ). The Mellin ransform of f (y) exiss for any v C on a1 R( v) a where f ( y) is defined as O y f ( y) O y a1 ( ), y a ( ), y Conversely, he Mellin inversion formula of (1) is defined as a j 1 M 1 ( F( v)) f (( y), v) F( v) j a j y v dv () where he inegraion is along a verical line hrough Re( v) a. Some of he basic fundamenal properies of he Mellin ransforms are deailed below. (a) Shifing Propery a a v1 M y f ( y), v y f ( y) y dy F ( v a) (3) (b) Scaling Propery v1 v M f ( ay), v f ( ay) y dy a F( v) (4) 484

3 Fadugba e al.; BJMCS, 9(6): , 15; Aricle no.bjmcs (c) The Mellin Transform of Derivaives d M f ( y), v ( 1) ( v ) F( v ), ( v ) V f, Z dy (5) where he symbol ( v ) is defined for ineger by; ( ) ( v )( v 1)...( v 1) v Some imporan resuls of he Mellin ransforms are given by d M v f ( y), v ( 1) v F( v) dy d M v f ( y), v ( 1) ( v ) F( v) dv d M v f ( y), v ( 1) ( v) F( v) dv (6) where Z and v( v 1)...( v 1) v. 3 New Technique for he Soluion of he Blac-Scholes Equaion for Vanilla Pu Opions Le us consider he Blac-Scholes equaion for vanilla pu opions given by P( S, ) S P( S, ) P( S, ) rs (, ), (, ), [, ] rp S S T S S (7a) wih he boundary condiions P( S, T ) max S,, on [, ) lim P( S, ) exp( r( T )), on [, T ) S lim P( S, ), on [, T ) S (7b) where is he volailiy, r is he ris-free ineres rae, is he srie price and T is he mauriy dae. I is a nown fac ha (7a) has a closed form soluion obained afer several change of variables and solving cerain relaed diffusion equaions. This procedure is no applicable in he vecor framewor where P( S, ) is a vecor and, r are marices. 485

4 Fadugba e al.; BJMCS, 9(6): , 15; Aricle no.bjmcs The Mellin ransform for he price of vanilla pu opions is defined as 1 M ( P( S, ), v) p( v, ) P( S, ) S v ds (8) The Mellin inversion formula of (8) is given by v M ( p( v, )) P(( S, ), v) p( v, ) S dv j (9) 1 1 a j a j To use he Mellin ransform and he condiions ha guaranee is exisence, we assume ha p( S, ) is bounded of polynomial degree when S and S i.e. O S p( S, ) O S a1 ( ), S a ( ), S (1) a R( v) a where a, ) for any vc on 1 ( 1 a is called fundamenal srip. Taing he Mellin ransform of he Blac-Scholes equaion for vanilla pu opion given by (7a), we have ha P( S, ) S P( S, ) P( S, ) M rs M rp( S, ) S S (11) Using he properies of he Mellin ransforms, we have P( S, ) d M p( v, ) d S P( S, ) M ( ) (, ) v v p v S P( S, ) M rs rvp( v, ) S M rp( S, ) rp( v, ) (1) Subsiuing (1) ino (11) and simplifying furher yields dp( v, ) v r v r p( v, ), [, T ] d (13) Inegraing (13) and solving furher yields 486

5 Fadugba e al.; BJMCS, 9(6): , 15; Aricle no.bjmcs v p( v, ) A( v)exp r v r (14) v Le ( v) r v r, hen (14) becomes p( v, ) A( v)exp ( v) (15) where A(v) is a consan of inegraion denoed by A( v) ( v, ) exp( ( v) T ) (16) ( v, ) is obained by aing he Mellin ransform of he iniial condiion given by hen we have P( S, T) ( S, v) max S, (17) v1 v1 (18) v( v 1) ( v, ) max( S,) S ds Using equaions (15), (16) and (18), we have ha v1 p( v, ) exp( ( v)( T )) v( v 1) (19) The Mellin inversion of (19) is obained as a j v1 1 1 v M ( p( v, )) P( S, ) exp( ( v)( T)) S dv j v( v 1) () a j Nex shall show ha () is a soluion of he Blac-Scholes equaion for vanilla pu opions given by (7a). Following he procedures in [4], we assume ha v m dv jdn, where v C (1) where m and n are he real and he imaginary pars of he complex number v C. Subsiuing (1) ino () yields 1m 1 ( m ) (, ) exp( ( )( )) ( m )( m 1) P S S m T dn () 487

6 Fadugba e al.; BJMCS, 9(6): , 15; Aricle no.bjmcs Subsiuing v m v ino( v) r v r, hen ( m ) ( m ) r ( m ) r m n m n rm r j mn rn (3) Since P( S, ) is coninuous and Mellin ransformable, seing T hen () becomes 1m ( m ) (4) m m 1 P( S, ) S dn ( )( 1) Equaion () saisfies (4) and i is well defined. Using he definiion of he Mellin ransforms, hen we have m1 M ( m) f ( s) s ds m R 1m ( m )( m 1) (5) and for [, T ) we have ha 1m ( m ) S exp( ( m )( T )) dn ( m )( m 1) m m m n M ( m) ST exp rm r ( T) exp ( T ) dn (6) Using he differeniaion heorem of parameric inegrals and he fac ha j n n exp ( T ) dn, j,1,,..., [, T ) (7) Then i follows ha upon differeniaion of (), one ges 1m P( S, ) 1 ( m ) ( m ) S exp( ( m )( T )) dn ( m )( m 1) 1m (8) P( S, ) 1 ( m 1) ( m ) S exp( ( m )( T )) dn S ( m )( m 1) 488

7 Fadugba e al.; BJMCS, 9(6): , 15; Aricle no.bjmcs m P( S, ) 1 S ( m ) ( m )( m 1) S exp( ( m )( T)) dn ( m )( m 1) (9) Subsiuing (), (8) and (9) ino (7a), we have ha P( S, ) S P( S, ) P( S, ) rs (, ) rec S S S 1m 1 ( m ) ( m ) S exp( ( m )( T )) dn ( m )( m 1) 1m S 1 ( m ) ( m )( m 1) S exp( ( m )( T )) dn ( m )( m 1) 1m 1 ( m 1) rs ( m ) S exp( ( m )( T )) dn ( m )( m 1) 1m 1 ( m ) r S exp( ( m )( T )) dn ( m )( m 1) 1 S ( ) ( 1) ( m ) ( m )( m 1) S rs ( m ) S r 1m ( m ) S exp( ( m )( T )) dn ( m )( m 1) 1 m n m n ( m ) rm r j mn rn 1m ( m ) S exp( ( m )( T )) dn ( m )( m 1) 1 m 1 ( m ) ( m ) ( m j ) exp( ( )( )) n S m T dn ( m )( m 1) Hence he expression for he price of vanilla pu opions denoed by P( S, ) and given by () is a soluion of (7a). 4 Conclusion In his paper we presened a new echnique for solving Blac-Scholes equaion for vanilla pu opions using he Mellin ransform mehod. The Mellin ransform mehod is a powerful echnique ha can also be used for (3) 489

8 Fadugba e al.; BJMCS, 9(6): , 15; Aricle no.bjmcs he valuaion of more complex vanilla opions and some pah dependen opions and can be exended o jump diffusion processes, sochasic volailiy and oher sochasic processes. Compeing Ineress Auhors have declared ha no compeing ineress exis. References [1] American Opion. Available: [] Blac F, Scholes M. The pricing of opions and corporae liabiliies. Journal of Poliical Economy. 1973;81: [3] Meron R. Theory of raional opion pricing. Bell Journal of Econom Managemen Science. 1973;4: [4] Fadugba SE, Ajayi AO. Alernaive approach for he soluion of he blac-scholes parial differenial equaion for European call opion. Open Access Library Journal. 13;:e146. [5] Panini R, Srivasav RP. Opion pricing wih Mellin ransforms. Mahemaical and Compuer Modelling. 4;4: (DOI:1.116/j.mcm.4.7.8) [6] AlAzemi F, AlAzemi A, Boyadjiev I. Mellin ransform mehod for solving he Blac-Scholes equaion. Inernaional Journal of Pure and Applied Mahemaics. 14;97: [7] J dar L, Sevilla P, Cores JC, Sala R. A new direc mehod for solving he Blac-Scholes equaion. Applied Mahemaics Leers. 5;18:9-3. [8] Sneddon I. The use of inegral ransforms. McGraw-Hill Boo Company; 197. [9] Yu. A. Brychov, Glaese HJ, Prudniov AP, Vu im Tuan. Mulidimensional inegral ransformaions. Gordon, Breach Science Publishers; 199. [1] Rodrigo MR, Mamon RS. An applicaion of Mellin ransform echniques o a Blac-Scholes equaion problem. Anal. Appl. 7;5: [11] Panini R, Srivasav RP. Pricing perpeual opions using Mellin ransforms. Applied Mahemaics Leers. 4;18: [1] Rodrigo M, Mamon R. An alernaive approach o solving he Blac Scholes equaion wih imevarying parameers. Applied Mahemaics Leers. 5;19: [13] Pearson ND. An efficien approach for pricing spread opions. Journal of Derivaives. 1995;3: [14] Carr P, Fague D. Valuing finie-lived opions as perpeual. Woring Paper; Available:hp://ssrn.com/absrac=76 [15] Mcean H. Appendix: A free boundary problem for he hea equaion arising from a problem in mahemaical economics. Indusrial Managemen Review. 1965;6:

9 Fadugba e al.; BJMCS, 9(6): , 15; Aricle no.bjmcs [16] Gradsheyn I, Ryzhi I. Table of inegrals, series, and producs, 7 h ediion. Academic Press; 7. [17] Carr P, Lee R. Pu-call symmery: Exensions and applicaions. Mahemaical Finance. 9;19: Fadugba e al.; This is an Open Access aricle disribued under he erms of he Creaive Commons Aribuion License (hp://creaivecommons.org/licenses/by/4.), which permis unresriced use, disribuion, and reproducion in any medium, provided he original wor is properly cied. Peer-review hisory: The peer review hisory for his paper can be accessed here (Please copy pase he oal lin in your browser address bar) hp://sciencedomain.org/review-hisory/