A Le Symmetry Analyss of the Black-Scholes Merton Fnance Model through modfed Local one-parameter transformatons by Tshdso Phanuel Masebe Submtted n accordance wth the requrements for the degree of Doctor of Phlosophy n the subject Appled Mathematcs at the Unversty of South Afrca September 2014 Supervsor : Dr J.M. Manale
Contents Introducton 1 1 Local One-parameter Pont Symmetres 4 1.1 Local One-parameter Pont transformatons.............. 4 1.2 Local One-parameter Pont transformaton groups........... 5 1.3 The group generator........................... 7 1.4 Prolongatons formulas......................... 7 1.4.1 The case N = 2, wth x 1 = x and x 2 = y............ 8 1.4.2 Invarant functons n R 2..................... 10 1.4.3 Mult-dmensonal cases..................... 11 1.4.4 Invarant functons n R N.................... 13 1.5 Determnng equatons.......................... 14 1.6 Le algebras................................ 14 1.7 Solvable Le algebras........................... 15 1.8 Le equatons............................... 16 1.9 Canoncal Parameter........................... 17 1.10 Canoncal varables............................ 17 1.11 One Dependent and Two Independent Varables............ 21
1.12 One Dependent and Three Independent Varables............ 22 1.13 One Dependent and Four Independent Varables............ 25 1.14 One Dependent and n Independent Varables.............. 26 1.15 m Dependent and n Independent Varables............... 27 2 Symmetry Analyss of Black-Scholes Equaton 30 2.1 Symmetres................................ 30 2.1.1 Fnte symmetry transformatons................ 32 2.2 Transformaton to heat equaton..................... 33 2.3 Invarant Solutons............................ 35 3 Introducton of new method 37 3.1 The transformed one-dmensonal Black-Scholes equaton....... 37 3.2 Soluton of determnng equaton for (3.6............... 39 3.2.1 Infntesmals for equaton (3.6................. 44 3.2.2 The symmetres for equaton (3.6................ 45 3.2.3 Table of Commutators...................... 46 3.3 Invarant Soluton for equaton (3.6................... 47 3.3.1 Solutons for equaton 3.97.................... 51 3.4 Two-Dmensonal Black-Scholes Equaton................ 54 3.4.1 Transformed two-dmensonal Black-Scholes Equaton..... 54 3.4.2 Soluton for determnng equaton for (3.109.......... 55 3.4.3 Infntesmals for equaton (3.109................ 63 3.4.4 Symmetres for equaton (3.109................. 66 3.4.5 Invarant Soluton for equaton (3.109............. 70
4 Transformaton of equaton (3.6 to heat equaton 74 4.1 Transformaton.............................. 75 5 Further Applcatons 80 5.1 Gaussan type partal dfferental equaton............... 80 5.1.1 Soluton of determnng equaton................ 81 5.1.2 Infntesmals........................... 84 5.1.3 Symmetres............................ 84 5.1.4 Invarant Solutons........................ 85 5.2 Symmetres n the epdemology of HIV and AIDS........... 89 5.3 Integratng (5.75............................. 91 5.3.1 Applyng the Le symmetry generator to (5.78........ 91 5.3.2 Infntesmals........................... 95 5.3.3 The symmetres.......................... 95 5.3.4 Constructon of nvarant solutons............... 96 Concluson 100 Appendces Appendx A: Manale s formulas and the nfntesmal ω........... Appendx B: Useful lmt results........................ v Appendx C: Soluton for determnng equaton (2.5............ v
Lst of Fgures 3.1 Commutator Table............................ 47 3.2 Plot for the soluton (3.100 wth dependent varable u represented on the vertcal axs and ndependent varable t represented on the horzontal axs.............................. 51 3.3 Plot for soluton (3.102 wth dependent varable u represented on the vertcal axs and ndependent varable λ represented on the horzontal axs.................................... 52 3.4 Plot for soluton (3.107 wth dependent varable u represented on the vertcal axs and ndependent varable r represented on the horzontal axs.................................... 54 3.5 Plot for soluton (3.226wth dependent varable u represented on the vertcal axs and ndependent varable r represented on the horzontal axs.................................... 73 v
Abstract The thess presents a new method of Symmetry Analyss of the Black-Scholes Merton Fnance Model through modfed Local one-parameter transformatons. We determne the symmetres of both the one-dmensonal and two-dmensonal Black-Scholes equatons through a method that nvolves the lmt of nfntesmal ω as t approaches zero. The method s dealt wth extensvely n [23]. We further determne an nvarant soluton usng one of the symmetres n each case. We determne the transformaton of the Black-Scholes equaton to heat equaton through Le equvalence transformatons. Further applcatons where the method s successfully appled nclude workng out symmetres of both a Gaussan type partal dfferental equaton and that of a dfferental equaton model of epdemology of HIV and AIDS. We use the new method to determne the symmetres and calculate nvarant solutons for operators provdng them. Keywords: Black-Scholes equaton; Partal dfferental equaton; Le Pont Symmetry; Le equvalence transformaton; Invarant soluton. v
Declaraton Student number: 46640193 I declare that A Le Symmetry Analyss of the Black-Scholes Merton Fnance Model through modfed Local one-parameter transformatons s my own work and that all the sources that I have used or quoted have been ndcated and acknowledged by means of a complete lst of references. SIGNAT URE DAT E v
Dedcaton I would lke to dedcate ths thess to my father Phlp Masebe, my late mother, Dolly Dasy Masebe who has always been my pllar of strength and an encouragement me to work hard n my studes. I further would lke to thank my son, Reotsheple Phllp Masebe who has always been a source of nspraton and motvaton for me to hold steadfast. v
Acknowledgement I am ndebted to my supervsor, Dr J.M. Manale for the gudance, patence and professonal assstance he has offered throughout ths thess. It s wth grattude that I have tapped from hs vast wsdom and knowledge. Specal words of thank you go to Prof M. T. Kambule for havng ntroduced me to Symmetry Analyss. I also acknowledge the teachng I receved from Prof N.H. Ibragmov n my Masters degree programme. It has rounded my understandng of Le Symmetry Analyss. I also thank Dr M.P. Mulaudz for the lessons he offered me on Mathematcs of Fnance. Words of thank you go to all my frends, former colleagues and current colleagues at Tshwane Unversty of Technology, and to Staff members n the Mathematcs Department at Unversty of South Afrca. Famly members, brothers and a sster wth whom we have a formdable bond I say thanks. Thanks to a specal lady for her words of encouragement, and expressng confdence n my ablty to put together an undertakng of ths nature. I acknowledge the fnancal support of my nsttuton Tshwane Unversty of Technology and the fnancal ad at Unversty of South Afrca. Wth God everythng s possble. v
Introducton The Le group analyss of dfferental equatons s the area of mathematcs poneered by Sophus Le n the 19th century (1849-1899. Sophus Le made a profound and far-reachng dscovery that all ad-hoc technques desgned to solve ordnary dfferental equatons (e.g separable varables, homogeneous, exact could be explaned and deduced smply by hs theory. These technques were n fact specal cases of a general ntegraton procedure of classfyng ordnary dfferental equatons n terms of ther symmetry groups. Ths dscovery led to Le dentfyng a full set of equatons whch could be ntegrated or reduced to lower order equatons by hs method [6],[14]. The Le Symmetry method s analytc and hghly algorthmc. The method systematcally unfes and extends ad-hoc technques to construct explct solutons for dfferental equatons. The emphass s on explct computatonal algorthms to dscover symmetres admtted by dfferental equatons and to construct nvarant solutons resultng from the symmetres [6],[13]. Le group analyss establshed tself to be an effectve method of solvng non-lnear dfferental equatons analytcally. In fact the frst general soluton of the problem of classfcaton was gven by Sophus Le for an extensve class of partal dfferental equatons [14]. Snce then many researchers have done work on varous famles of dfferental equatons. The results of ther work have been captured n several outstandng lterary works by amongst others Ibragmov (1999, Hydon(2000, Bluman and Anco(2002, etc [6], [9], [13]. 1
The present project ttled A Le Symmetry Analyss of the Black-Scholes Merton Fnance Model through modfed Local one-parameter transformatons seeks to explore the analyss of the wdely used one-dmensonal model of Black-Scholes partal dfferental equaton through determnng the new symmetres and nvarant solutons of some of these symmetres. The analyss s through a new method developed n [23], [28]. Further exploraton of the method can be found n [25], [26],[27]. Throughout the project we use Le pont symmetres. The Black-Scholes equaton s a partal dfferental equaton that governs the value of fnancal dervatves. Determnng the value of dervatves had been a problem n fnance for almost 70 years snce 1900. In the early 70s, Black and Scholes made a poneerng contrbuton to fnance by developng a Black-Scholes equaton under restrctve assumptons and the opton valuaton formula. Scholes obtaned a Nobel Prze for economcs n 1997 for hs contrbuton (Black had passed on n 1995 and could not receve the prze personally [32]. Furthermore, nformaton on the dervaton of and the restrctve assumptons n the development of the Black-Scholes equaton can be obtaned n the texts [4], [8], [10], [11], [12], [21], [22] and [30]. The thess outlne s as follows: Chapter 1 presents the concept of Local One-parameter Pont Symmetres. We ntroduce concepts of Local One-parameter Pont transformatons, generator, prolongaton formulas, determnng equaton and Le algebras. These concepts serve as tools n the analyss of the Black-Scholes partal dfferental equaton. Chapter 2 presents the Symmetry Analyss of equaton (2.1 as outlned n [9]. The chapter presents the symmetry of (2.1,fnte symmetry transformatons and the transformaton of the equaton (2.1 to the heat equaton. 2
Chapter 3 s the core of the thess. It ntroduces the Symmetry Analyss of equaton (2.1 by usng the method developed through contrbutons n ([23] and ([28]. The chapter detals the symmetres of one-dmensonal Black-Scholes equaton (2.1 as well as the two-dmensonal case through modfed Local one-parameter transformatons. Chapter 4 nvolves the transformaton of the transformed Black-Scholes equaton to heat through Le equvalence transformatons. We calculate ths transformaton by usng a method smlar but wth slght modfcaton to that developed n [17]. Chapter 5 presents other areas where the method was successfully appled. The applcaton ncludes determnng the symmetres and nvarant solutons of the Gaussan type dfferental equaton and symmetres n the epdemology of HIV and AIDS. Contrbutons to the study The thess s based on peer revewed contrbutons to the study as outlned n the publcatons ([1] and ([28]. Several texts for example ([6], ([13],([14] etc. present symmetres of dfferental equatons n the way ntally ntroduced by Sophus Le. However, n ths publcatons we ntroduce an alternatve way to determne these symmetres, and the method present new addtonal symmetres. 3
Chapter 1 Local One-parameter Pont Symmetres The chapter presents the underlyng theory of Le Symmetry Analyss and the tools we wll use n subsequent chapters. The most common of all symmetres n practce are Local One-parameter Pont symmetres. 1.1 Local One-parameter Pont transformatons To begn, we consder the followng defnton. Defnton 1 Let x = G (x; ɛ (1.1 be a famly of one-parameter ɛ R nvertble transformatons, of ponts x = (x 1,, x N R N nto ponts x = ( x 1,, x N R N. These are known as oneparameter transformatons, and subject to the condtons x ɛ=0 = x. (1.2 That s, G (x; ɛ = x. (1.3 ɛ=0 4
Expandng (1.1 about ɛ = 0, n some neghborhood of ɛ = 0, we get ( ( ( G x = x + ɛ + ɛ2 2 G G + = x + ɛ + O ( ɛ 2. (1.4 ɛ 2 ɛ 2 ɛ ɛ=0 ɛ=0 ɛ=0 Lettng reduces the expanson to ξ (x = G ɛ, (1.5 ɛ=0 x = x + ɛξ (x + O ( ɛ 2. (1.6 Defnton 2 The expresson s called a Local One-parameter Pont transformaton. x = x + ɛξ (x, (1.7 The components of ξ (x are called the nfntesmals of (1.1 [6]. 1.2 Local One-parameter Pont transformaton groups The set G of transformatons ( G x ɛ = x + ɛ ɛ ɛ =0 + ɛ2 2 ( 2 G ɛ 2 ɛ =0 becomes a group only when truncated at O (ɛ 2. +, = 1, 2, 3,, (1.8 That s, G s a group snce the followng propertes hold under bnary operaton +: 1. Closure. If x ɛ1, x ɛ2 G and ɛ 1, ɛ 2 R, then x ɛ1 + x ɛ2 = (ɛ 1 + ɛ 2 ξ (x = x ɛ3 G, and ɛ 3 = ɛ 1 + ɛ 2 R. 2. Identty. If x 0 I G such that for any ɛ R then x 0 s an dentty n G. x 0 + x ɛ = x ɛ = x ɛ + x 0, 5
3. Inverses. For x ɛ G, ɛ R, there exsts x 1 ɛ G, such that x 1 ɛ + x ɛ = x ɛ + x 1 ɛ, x 1 ɛ = x ɛ 1, and ɛ 1 = ɛ D, where + s a bnary composton of transformatons and t s understood that x ɛ = x ɛ x. Assocatvty follows from the closure property. Example 1 : Group of Rotatons n the Plane x 1 = x 1 cos ɛ + x 2 sn ɛ, x 2 = x 2 cos ɛ x 1 sn ɛ. That s, x 1 = x 1 + x 2 ɛ, x 2 = x 2 x 1 ɛ. Example 2 : Group of Translatons n the Plane x = x + ɛ, x j = x j. Example 3 : Group of Scalngs n the Plane 6
x = (1 + ɛx, x j = (1 + ɛ 2 x j. [6], [15]. 1.3 The group generator The Local One-parameter Pont transformatons n (1.7 can be rewrtten n the form x = x + ɛξ (x x, (1.9 so that An operator, x = (1 + ɛξ (x x. (1.10 G = ξ (x, (1.11 can then be ntroduced, so that (1.9 assumes the form x = (1 + ɛg x. (1.12 The operator (1.11 has the expanded form G = N k=1 ξ k x, (1.13 k or smply G = ξ k. [6], [16], [18]. (1.14 xk 1.4 Prolongatons formulas It often happens that the nvarant functon F does not only depend on the pont x alone, but also on the dervatves. When that s the case then we have to use the prolonged form of the operator G. 7
1.4.1 The case N = 2, wth x 1 = x and x 2 = y The case N = 2, wth x 1 = x and x 2 = y reduces (1.13 to G = ξ(x, y x + η(x, y y. (1.15 In determnng the prolongatons, t s convenent to use the operator of total dfferentaton where D = x + y y + y y +, (1.16 y = dy dx, The dervatves of the transformed pont s then y = d2 y dx 2,. (1.17 Snce ȳ = dȳ d x. (1.18 x = x + ɛξ and ȳ = y + ɛη, (1.19 then That s, ȳ = dy + ɛdη dx + ɛdξ. (1.20 ȳ = Now ntroducng the operator D: dy/dx + ɛdη/dx dx/dx + ɛdξ/dx. (1.21 Hence ȳ = y + ɛd(η 1 + ɛd(ξ = (y + ɛd(η(1 ɛd(ξ 1 ɛ 2 (D(ξ 2. (1.22 8
That s, ȳ = y ɛ(d(η y D(ξ ɛ 2 (D(ξ 1 ɛ 2 (D(ξ 2. (1.23 ȳ = y + ɛ(d(η y D(ξ, (1.24 or ȳ = y + ɛζ 1, (1.25 wth ζ 1 = D(η y D(ξ. (1.26 It expands nto The frst prolongaton of G s then ζ 1 = η x + (η y ξ x y y 2 ξ y. (1.27 G [1] = ξ(x, y x + η(x, y y + ζ1 y. (1.28 For the second prolongaton, we have wth ȳ = y + ɛd(ζ 1 1 + ɛd(ξ y + ɛζ 2, (1.29 ζ 2 = D(ζ 1 y D(ξ. (1.30 Ths expands nto ζ 2 = η xx + (2η xy ξ xx y + (η yy 2ξ xy y 2 y 3 ξ yy + (η y 2ξ x 3y ξ y y. (1.31 The second prolongaton of G s then 9
G [2] = ξ(x, y x + η(x, y y + ζ1 y + ζ2. (1.32 y Most applcatons nvolve up to second order dervatves. It s reasonable then to pause here, for ths case [14]. 1.4.2 Invarant functons n R 2 Theorem 1 A functon F (x, y s an nvarant of the group of transformatons (1.13 f for each pont (x, y t s constant along the trajectory determned by the totalty of transformed ponts ( x, ȳ: F ( x, ȳ = F (x, y. (1.33 Ths requres that GF = 0, (1.34 leadng to the characterstc system dx ξ = dy η. (1.35 Proof. Consder the Taylor seres expanson of F ( x wth respect to ɛ: F ( x, ȳ = F ( x, ȳ + ɛ F +. (1.36 ɛ=0 ɛ ɛ=0 Ths can be wrtten n the form ( F ( x, ȳ = F ( x, ȳ x + ɛ F ɛ=0 ɛ x + ȳ F +. (1.37 ɛ ȳ ɛ=0 That s, or Hence ( F ( x, ȳ = F (x, y + ɛ ξ F x + η F +, (1.38 ȳ ɛ=0 ( F ( x, ȳ = F (x, y + ɛ ξ x + η y F +. (1.39 F ( x, ȳ = F (x, y + ɛg F, (1.40 10
wth For ɛ = 0 then we get G = ξ x + η y. (1.41 F ( x, ȳ = F (x, y, (1.42 thus provng the theorem [16]. 1.4.3 Mult-dmensonal cases In dealng wth the mult-dmensonal cases, we may recast the generator (1.13 as We consder the kth-order partal dfferental equaton G = ξ x + ηα u. (1.43 α F (x, u, u (1, u (2,..., u (k where x = (x 1... x n, u (1 = u x. (1.44 By defnton of symmetry, the transformatons (1.1 form a symmetry group G of the system (1.44 f the functon ū = ū( x satsfes (1.43 whenever the functon u = u(x satsfes (1.44. The transformed dervatves ū (1,..., ū (k are found from (1.4 by usng the formulae of change of varables n the dervatves, D = D (f j D j. [6] Here D = x + ua ( u + a ua j( + (1.45 u a j s the total dervatve operator w.r.t. x and D j s gven n terms of the transformed varables. The transformatons (1.13 together wth the transformatons on ū (1 form a group, G [1], whch s the frst prolonged group whch acts n the space (x, u, ū (1. Lkewse, we obtan the prolonged groups G [2] and so on up to G [k]. The nfntesmal transformatons of the prolonged groups are: ū a u a + aζ a (x, u, u (1, ū a j u a j + aζ a j(x, u, u (1, u (2,. ū a 1... k u a 1... k + aζ a 1... k (x, u, u (1,..., u (k. (1.46 11
The functons ζ a (x, u, u (1, ζ a j(x, u, u (1, u (2 and ζ a 1... k (x, u, u (1,..., u (k are gven, recursvely, by the prolongaton formulas: ζ a = D (η a u a j D (ξ j, ζ a j = D j (ζ a u l D j (ξ l,. (1.47 ζ a 1... k = D k (ζ a 1... k 1 u l1... k 1 D k (ξ l. The generator of the prolonged groups are: G [1] = ξ (x, u x + η a (x, u u a + ζ a (x, u, u (1 u a. (1.48, G [k] = ξ (x, u x + η a (x, u u a + ζ a (x, u, u (1 u a + + ζ a 1... k (x, u,..., u (k u a 1... k. [6], [15]. (1.49 Defnton 3 A dfferental functon F (x, u,..., u (p, p 0, s a pth-order dfferental nvarant of a group G f F (x, u,..., u (p = F ( x, ū,..., ū (p, (1.50.e. f F s nvarant under the prolonged group G [p], where for p = 0, u (0 u and G [0] G. Theorem 2 A dfferental functon F (x, u,..., u (p, p 0, s a pth-order dfferental nvarant of a group G f G [p] F = 0, (1.51 where G [p] s the pth prolongaton of G and for p = 0, G [0] G. The substtuton of (1.49 and (1.50 nto (1.5 gves rse to E σ ( x, ū, ū (1,..., ū (k E σ (x, u, u (1,..., u (k + a(g [k] E σ, (1.52 σ = 1,..., m. 12
Thus, we have G [k] E σ (x, u, u (1,..., u (k = 0, σ = 1,..., m, (1.53 whenever (1.44 s satsfed. The converse also apples. 1.4.4 Invarant functons n R N Theorem 3 A functon F (x s an nvarant of the group of transformatons (1.13 f for each pont x t s constant along the trajectory determned by the totalty of transformed ponts x: F ( x = F (x. (1.54 Ths requres that GF = 0, (1.55 leadng to the characterstc system dx 1 ξ 1 = = dxn ξ N. (1.56 Proof. Consder the Taylor seres expanson of F ( x wth respect to ɛ: F ( x = F (x + ɛ F +. (1.57 ɛ=0 ɛ ɛ=0 Ths can be wrtten n the form F ( x = F ( x + ɛ x ɛ=0 ɛ F +. (1.58 ɛ=0 That s, For ɛ = 0 then we get F ( x = F ( x + ɛξ F +. ɛ=0 ɛ=0 (1.59 F ( x = F (x, (1.60 thus provng the theorem [16]. 13
1.5 Determnng equatons Equatons (1.53 are called the determnng equatons. They are wrtten compactly as G [k] E σ (x, u, u (1,..., u (k (1 = 0, σ = 1,..., m, where (1 means evaluated on the surface (1.44. The determnng equatons are lnear homogeneous partal dfferental equatons of order k for the unknown functons ξ (x, u and η a (x, u. These are consequences of the prolongaton formulae (1.47. Equatons (1.5 also nvolve the dervatves u (1,..., u (k some of whch are elmnated by the system (1.44. We then equates the coeffcents of the remanng unconstraned partal dervatves of u to zero. In general, (1.5 decomposes nto an overdetermned system of equatons, that s, there are more equatons than the n + m unknowns ξ and η a. There are computer algebra programs that can perform the task of solvng determnng equatons [3]. Snce the determnng equatons are lnear homogeneous, ther solutons form a vector space L [6]. 1.6 Le algebras There s another mportant property of the determnng equatons, vz. f the generators and G 1 = ξ 1(x, u x + ηa 1(x, u u a G 2 = ξ 2(x, u x + ηa 2(x, u u a satsfy the determnng equatons, so do ther commutator [G 1, G 2 ] = G 1 G 2 G 2 G 1 [G 1, G 2 ] = ( G 1 (ξ2 G 2 (ξ1 x + (G 1(η2 a G 2 (η1 a u a whch obeys the propertes of blnearty, skew-symmetry and Jacob s dentty, vz. 14
1. Blnearty. If G 1, G 2, G 3 L, then [αg 1 + βg 2, G 3 ] = α [G 1, G 3 ] + β [G 2, G 3 ], α, β are scalars 2. Skew symmetry. If G 1, G 2 L, then [G 1, G 2 ] = [G 2, G 1 ]. 3. Jacob Identty. If G 1, G 2, G 3 L, then [[G 1, G 2 ], G 3 ] + [[G 2, G 3 ], G 1 ] + [[G 3, G 1 ], G 2 ] = 0. The vector space L of all solutons of the determnng equatons forms a Le algebra whch generates a mult-parameter group admtted by (1.44 [15],[16]. 1.7 Solvable Le algebras In ths secton, we wll show that f r = 1, then the order of an ODE can be reduced constructvely by one. If n > 2 and r = 2, the order can be reduced constructvely by two. But f n > 2 and r > 2, t wll not necessarly follow that the order can be reduced by more than one. However, f the r-dmensonal Le algebra of nfntesmal generators of an admtted r-parameter group has a q-dmensonal solvable subalgebra, then the order of the ODE can be reduced constructvely by q. Defnton 4 A subalgebra A of the Le algebra L r wth dmenson r, s called an deal or normal subalgebra of L r f [G α, G β ] A, for all G α A and G β L r. Defnton 5 The Le algebra L r, wth dmenson r, s called an r-dmensonal solvable Le algebra f there exsts a chan of subalgebras A 1 A 2 L r, wth A k 1 beng an deal of A k, and k r. 15
Defnton 6 A Le algebra A s Abelan f [G α, G β ] = 0, f both G α and G β are n A. [16] Theorem 4 An abelan algebra s solvable [16]. Theorem 5 A two-dmensonal algebra s solvable. Proof Let L be a two-dmensonal Le Algebra wth nfntesmal generators X 1 and X 2 as bass vectors. Suppose that [X 1, X 2 ] = ax 1 + bx 2 = Y If C 1 X 1 + C 2 X 2 L, then [Y, C 1 X 1 + C 2 X 2 ] = C 1 [Y, X 1 ] + C 2 [Y, X 2 ] = C 1 b[x 2, X 1 ] + C 2 a[x 1, X 2 ] = (C 2 a C 1 by Therefore Y s a one-dmensonal deal of L. Hence the proof [5]. 1.8 Le equatons One-parameter groups are obtaned by ther generators by means of Le s theorem: Theorem 6 Gven the nfntesmal transformatons x = x + ɛξ (x, u α = u α + ɛη α (x or ts symbol G, the correspondng one-parameter group G s obtaned by soluton of the Le equatons d x dɛ = ξ ( x, ū, dū α dɛ = ηα ( x, ū, subject to the ntal condtons x ɛ=0 = x, ū α ɛ=0 = u α. [18] (1.61 16
1.9 Canoncal Parameter If n the group property 1., dscussed above, the expresson ϕ(ɛ 1, ɛ 2 can be wrtten as ϕ(ɛ 1, ɛ 2 = ɛ 1 + ɛ 2, then the parameter a s sad to be canoncal. In general, a canoncal parameter exsts whenever ϕ exsts. That s, one has the followng theorem: Theorem 7 : For any ϕ(a, b, there exsts the canoncal parameter ã = a 0 da A(a, where A(a = ϕ(a, b b=0. [18] b Ths system, wth a as the canoncal parameter below, transforms form-nvarantly n varables t, x, y, z, u, v, w, p, µ (see [?] under where [ a t = t exp 0 [ a z = z exp da 0 µf (µ [ a w = w exp ] [ da a, x = x exp µf (µ 0 [ a 0 ], ū = u exp da µf (µ 0 ], p = p exp F (µ = ] [ da a, ȳ = y exp µf (µ 0 [ a da µf (µ [ a 0 1 µf (µ, [16] ], v = v exp da µf (µ 0 ] da, µf (µ da µf (µ ], F ( µ = a + F (µ, ], 1.10 Canoncal varables Theorem 8 : Every one-parameter group of transformatons ( x = f(x, y, ɛ, ȳ = g(x, y, ɛ reduces to a group of translatons t = t + ɛ, ū = u wth the generator X = t 17
by a sutable change of varables t = t(x, y, u = u(x, y. The varables t, u are called canoncal varables. Proof: Under change of varables the dfferental operator transforms accordng to the formula X = ξ(x, y x + η(x, y y X = X(t t + X(u u. (1.62 Therefore, canoncal varables are found from the lnear partal dfferental equaton of the frst order: Hence the proof [15]. t(x, y t(x, y X(t ξ(x, y + η(x, y x y X(u ξ(x, y u(x, y x + η(x, y u(x, y y = 1 = 0. (1.63 Theorem 9 : By a sutable choce of the bass G 1, G 2, any two-dmensonal Le algebra can be reduced to one of the four dfferent types, whch are determned by the followng canoncal structural relatons: I. II. III. IV. [G 1, G 2 ] = 0, G 1 G 2 0; (1.64 [G 1, G 2 ] = 0, G 1 G 2 = 0; (1.65 [G 1, G 2 ] 0, G 1 G 2 0; (1.66 [G 1, G 2 ] 0, G 1 G 2 = 0, (1.67 18
where G 1 G 2 = ξ 1 η 2 η 1 ξ 2, and G 1 = ξ 1 x + η 1 y, G 1 = ξ 2 x + η 2 y. [15]. Type I. [G 1, G 2 ] = 0, G 1 G 2 0. Ths condton reduces y = f(y, to wth C 1 beng the ntegraton constant. dy f(y = x + C 1, Type II. [G 1, G 2 ] = 0, G 1 G 2 = 0; Ths condton reduces y = f(x, to ( y = f(xdx dx + C 1 x + C 2. wth C 1 and C 2 beng the ntegraton constants. 19
Type III. [G 1, G 2 ] 0, G 1 G 2 0; Ths condton reduces y = 1 x f(y, to wth C 1 beng the ntegraton constant. dy f(y = ln(x + C 1, Type IV. [G 1, G 2 ] 0, G 1 G 2 = 0, Ths condton reduces y = y f(x, to y = C 1 e f(xdx dx + C 2. Theorem 10 : The bass of an algebra L r can be reduced by a sutable change of varable to one of the followng forms: I. II. G 1 = x, G 2 = y ; G 1 = y, G 2 = x y ; 20
III. IV. G 1 = y, G 2 = x x + y y ; G 1 = y, G 2 = y y. The varables x and y are called canoncal varables. 1.11 One Dependent and Two Independent Varables. We consder the equatons u t = u xx, (1.68. In order to generate pont symmetres for equaton (1.68, we frst consder a change of varables from t, x and u to t, x and u nvolvng an nfntesmal parameter ɛ. A Taylor s seres expanson n ɛ near ɛ = 0 yelds t t + ɛt (t, x, u x x + ɛξ(t, x, u ū u + ɛζ(t, x, u (1.69 where t ɛ ɛ=0 = T (t, x, u x ɛ ɛ=0 = ξ(t, x, u ū ɛ ɛ=0 = ζ(t, x, u The tangent vector feld (1.12 s assocated wth an operator. (1.70 G = T t + ξ x + ζ u, (1.71 called a symmetry generator. Ths n turn leads to the nvarance condton G [2] [F (t, x, u t, u x, u tx, u tt, u xx ] {F (t,x,ut,u x,u tx,u tt,u xx=0} = 0, (1.72 21
where G [2] s the second prolongaton of G. It s obtaned from the formulas: where and G [2] = G + ζt 1 u t + ζx 1 u x + ζtt 2 + ζtx 2 u tx + ζxx 2 u xx, u tt ζt 1 = g + u f + [ ] f T ut ξ t t t ζ 1 x = g x + u f x + [ f ξ x ζ 2 tt = 2 g t 2 [ + u 2 f + t 2 + [ f 2 T t [ ζxx 2 = 2 g + u 2 f + x 2 x 2 ζ 2 tx = + [ f 2 T x [ 2 g + u 2 f + t x t x [ + 2 f 2 ξ t t x ] 2 f 2 T t t 2 u x x, (1.73 ] ux T t t, (1.74 ] utt 2 ξ t u tx, ] 2 f 2 ξ x x 2 u t 2 ξ t 2 u x u x 2 T x 2 u t ] uxx 2 T x u tx, 2 f 2 T x t x ] u t ] u x [ f T t ξ x] utx T x u tt ξ t u xx. 1.12 One Dependent and Three Independent Varables. In order to generate pont symmetres for equaton (1.68, we frst consder a change of varables from t, x and u to t, x and u nvolvng an nfntesmal parameter ɛ. A Taylor s seres expanson n ɛ near ɛ = 0 yelds t t + ɛt (t, x, u x x + ɛξ(t, x, u (1.75 ū u + ɛζ(t, x, u where t ɛ ɛ=0 = T (t, x, u x ɛ ɛ=0 = ξ(t, x, u ū ɛ ɛ=0 = ζ(t, x, u. (1.76 22
The tangent vector feld (1.12 s assocated wth an operator G = T t + ξ x + ζ u, (1.77 called a symmetry generator. Ths n turn leads to the nvarance condton G [2] [F (t, x, u t, u x, u tx, u tt, u xx ] {F (t,x,ut,u x,u tx,u tt,u xx=0} = 0, (1.78 where G [2] s the second prolongaton of G. It s obtaned from the formulas: G [2] = G + ζt 1 u t + ζx 1 u x + ζtt 2 + ζtx 2 u tx + ζxx 2 u xx, u tt where ζt 1 = g + u f + [ ] f T ut ξ t t t x u x. (1.79 ζ 1 x = g x + u f x + [ f ξ x ] ux T t u t. (1.80 and ζ 2 tt = 2 g t 2 [ + u 2 f + t 2 + [ f 2 T t [ ζxx 2 = 2 g + u 2 f + x 2 x 2 ζ 2 tx = + [ f 2 T x 2 g + u 2 f t x [ + 2 f 2 ξ t t x ] 2 f 2 T t t 2 ] utt 2 ξ t u tx. ] 2 f 2 ξ x x 2 u t 2 ξ t 2 u x u x 2 T x 2 u t ] uxx 2 T x u tx. [ ] + 2 f 2 T u t x x t x t ] u x [ f T ξ t x] utx T x u tt ξ t u xx. We now look at two dmensonal and three dmensonal heat equaton gven respectvely by u t = u xx + u yy (1.81 and u t = u xx + u yy + u zz. (1.82 23
In order to generate pont symmetres for equaton (1.81, we frst consder a change of varables from t, x, y and u to t, x, y and u nvolvng an nfntesmal parameter ɛ. A Taylor s seres expanson n ɛ near ɛ = 0 yelds t t + ɛt (t, x, y, u, x x + ɛξ(t, x, y, u, (1.83 ȳ y + ɛϕ(t, x, y, u, ū u + ɛζ(t, x, y, u. where t ɛ ɛ=0 = T (t, x, y, u, x ɛ ɛ=0 = ξ(t, x, y, u, ȳ ɛ ɛ=0 = ϕ(t, x, y, u, ū ɛ ɛ=0 = ζ(t, x, y, u.. (1.84 The tangent vector feld (1.84 s assocated wth an operator G = T t + ξ x + ϕ y + ζ u, (1.85 called a symmetry generator. Ths n turn leads to the nvarance condton G [2] [F (t, x, u t, u x, u y, u tx, u ty, u tt, u xx, u xy, u yy ] {F (t,x,ut,u x,u y,u tx,u ty,u tt,u xx,u xy,u yy=0} = 0, where G [2] s the second prolongaton of G. It s obtaned from the formulas: G [2] = G + ζt 1 u t + ζx 1 u x + ζy 1 u y + ζtx 2 u tx + ζty 2 u ty + ζxx 2 u xx + ζxy 2 + ζ 2 tt u xy u tt + ζ 2 yy u yy, (1.86 where ζt 1 = g + u f + [ ] f T ut ξ u ϕ t t t x x u y t, (1.87 ζ 1 x = g x + u f x + [ f ξ x ] ux T t u t u y ϕ x, (1.88 ζy 1 = g + u f + [ ] f ϕ uy ϕ u ξ y y x t t u x u y t T, (1.89 y [ ζ 2 tt = 2 g t 2 + u 2 f t 2 + + [ f 2 T t u t 2 ξ u t 2 x 2 ξ u t 2 y ] utt 2 ξ u t tx 2 ϕ u t yt, ] 2 f 2 T t t 2 24
ζ 2 ty = ζ 2 tx = [ + 2 f 2 ξ t t x ζ 2 ty = [ ] + f 2 T u t x x t x t ] u x u 2 ϕ y [ 2f 2 T t x ] uxx [ 2 ϕ + ϕ t [ ] + 2 f 2 T u t x x t x t 2 g + u 2 f t x [ ξ t + ξ x 2 g + u 2 f t x [ + 2 f 2 ξ t t x ζ 2 xx = 2 g x 2 + u 2 f x 2 + 2 g t y + u 2 f [ 2 ϕ + ϕ t y [ + [ f 2 ξ x [ + t y ] u yy 2 [ ζyy 2 = 2 g + u 2 f + y 2 2 y [ f 2 ϕ y ] T utx t x x ] uxy. ] u x [ f T t ξ x] utx T x u tt ξ u t xx. ] 2 f 2 ξ x x 2 u x 2 ϕ x 2 u y 2 T x 2 u t ] uxx 2 ϕ u x xy, 2 T ] 2 f 2 ϕ y t y [ ξ t + ξ y ] 2 f 2 ϕ y y 2 ] u yy 2 [ ξ y ] u xy + x u tx, u y 2 ξ u t y x 2 T u y 2 t [ 2f 2 T T t y u y 2 ξ u y 2 x 2 T [ 2f 2 T y ] u xy + y 2 u t ] u yt, ] u yt, 1.13 One Dependent and Four Independent Varables. For the equaton (1.82 the tangent vector s gven by and G = T t + ξ x + ϕ y + β z + ζ u, (1.90 G [2] = G + ζt 1 u t + ζx 1 u x + ζy 1 u y + ζz 1 u z + ζtx 2 u tx + ζty 2 u ty + ζxx 2 u xx + ζxy 2 + ζ 2 zz u zz u xy + ζ 2 zt + ζ 2 xz u zt u xz + ζ 2 tt + ζ 2 yy u tt u yy + ζ 2 yz u yz ζzz 2 = 2 g + 2 f u + u z 2 y 2 z (2 f 2 β u 2 ξ z z 2 x u 2 ϕ z 2 y u 2 τ y 2 t z 2 2u zx ξ z 2u zy ϕ z 2u zt τ z, + u zz (f 2 β z ζnn 2 = 2 g n + 2 f 2 (n 1 u + u n(2 f 2 n 2 θ n + (1.91 2 25
1.14 One Dependent and n Independent Varables. The Local One-parameter Pont transformatons x = X (x, u, ɛ = x + ɛξ(x, u + 0ɛ 2 (1.92 ū = U(x, u, ɛ = u + ɛη(x, u + 0ɛ 2 = 1, 2,..., n (1.93 actng on (x, u - space has generator X = ξ (x, u x + η(x, u u The kth extended nfntesmals are gven by ξ(x, u, η(x, u, η (1 (x, u, u,..., η (1 (x, u, u,..., u (1, (1.94 and the correspondng kth extended generator s X (k = X +η x u +ζ1 u +...+ζ k u 1 2... l = 1, 2,..., n l = 1, 2,..., k k 1[6]. (1.95 Theorem 11 The extended nfntesmals satsfy the recursve relatons η (1 = D η (D ξ j u j, = 1, 2,..., n (1.96 ζ k 1 2... k = D k ζ k 1 1 2... k 1 (D ξ j u 1 2... k 1 j (1.97 = 1, 2,..., n for l = 1, 2,..., k wth k 2 Proof. Let A be an n n matrx D 1 X 1 D 1 X n A =.. D n X 1 D n X n and assume that A 1 exsts. From equaton (1.92 and the matrx A we have that D 1 (x 1 + ɛξ 1 D 1 (x 2 + ɛξ 2 D 1 (x n + ɛξ n D A = 2 (x 1 + ɛξ 1 D 2 (x 2 + ɛξ 2 D 2 (x 2 + ɛξ 2 + 0(ɛ 2 = I + ɛb + 0(ɛ 2.. D n (x 1 + ɛξ 1 D n (x 2 + ɛξ 2 D n (x n + ɛξ n 26
where I s the dentty matrx and D 1 ξ 1 D 1 ξ 2 D 1 ξ n D B = 2 ξ 1 D 2 ξ 2 D 2 ξ n... D n ξ 1 D n ξ 2 D n ξ n Then A 1 = I ɛb + 0(ɛ 2 Usng some transformatons we arrve at that ζ k 1 2... k 1 D 1 ζ k 1 1 2... u 1 2... k 1 1 ζ k 1 2... k 2 D = 2 ζ k 1 1 2... u B 1 2... k 1 2... ζ k 1 2... k n D n ζ k 1 1 2... u 1 2... k 1 n = 1, 2,..., n for l = 1, 2,..., k wth k 2 and ths leads to (1.97. Hence the proof. The detals of the proof are contaned n [6]. 1.15 m Dependent and n Independent Varables. We consder the case of n ndependent varables x = (x 1..... x n and m dependent varables u(x = u 1 (x... u n (x. Partal dervatves are denoted by u µ notaton u 1 u = u 1 1(x... u 1 n(x... u m 1 (x... u m n (x denotes the set of all frst-order partal dervatves = uµ x. The p u = {u p 1... p µ = 1... m : 1... p = 1... n} = { p u p (x x 1... x p µ = 1... m : 1... p = 1... n} denotes the set of all partal dervatves of order p. Pont transformatons of the form x = f(x, u (1.98 ū = g(x, u (1.99 27
actng on the n + m dmensonal space (x, u has as ts pth extended transformaton wth, 1,..., p = 1,..., n; µ = 1... m; (x = f (x, u (1.100 (u µ = g µ (x, u (1.101 (u µ = hµ (x, u, u (1.102. (1.103 (u µ 1... p = h µ 1... p (x, u, u... p u (1.104 (u µ ( (x the frst-order dervatves are determned by (u µ 1 h µ 1 D 1 g µ (u µ 2 h µ = 2 D = A 1 2 g µ... (u µ n h µ n D n g µ where A 1 s the nverse of the matrx D 1 f 1 D 1 f n A =.. D n f 1 D n f n n terms of the total dervatve operators = 1,..., n D = x + uµ u + µ uµ 1 u µ +..., 1. The transformed components of [7]. The transformed components of the hgher-order dervatves are determned by (u µ 1... p 1 (u µ 1... p 2. (u µ 1... p n = h µ 1... p 1 h µ 1... p 2. h µ 1... p n = A 1 D 1 h µ 1... p 1 1 D 2 h µ 1... p 1 2. D n h µ 1... p 1 n The stuaton where the pont transformaton (5.154,5.155 s a one-parameter group of transformaton gven by x = f (x, u, ɛ = x + ɛξ (x, u + 0(ɛ 2, = 1,...n (1.105 ū µ = g µ (x, u, ɛ = u µ + ɛξ (x, u + 0(ɛ 2, µ = 1,...m (1.106 28
wll have the correspondng generator gven by X = ξ (x, u x + ηµ (x, u u µ [7] (1.107 29
Chapter 2 Symmetry Analyss of Black-Scholes Equaton In ths chapter we state the Symmetry Analyss as presented n the paper by Ibragmov and Gazzov ([9]. We present the symmetres of one-dmensonal Black-scholes equaton, fnte symmetry transformatons of determned operators, transformaton to heat equaton and nvarant solutons from some operators. The one-dmensonal Black-scholes model s gven by the partal dfferental equaton u t + 1 2 A2 x 2 u xx + Bxu x Cu = 0 (2.1 wth constant coeffcents A, B and C, where A 0, and defne D B A2 2. The Symmetry Analyss of the equaton (2.1 as outlned n [9] follows n the subsequent sectons. 2.1 Symmetres For the Black-Scholes model (2.1, n = 1, x 1 = x the generator or symbol of nfntesmal symmetry s gven as X = ξ 1 (t, x t + ξ2 (t, x + η(t, x x u (2.2 30
Its extenson up to the second prolongaton s gven by X (2 = X + η (1 t + η (1 x + η (2 xx (2.3 u t u x u xx where X s defned by equaton (2.2 and the functons ζ 0, ζ 1 and ζ 11 are gven by ζ 0 = D t (η u t D t (ξ 0 u xd t (ξ 1 = η t + u t η u u t ξ 0 t u 2 t ξ 0 u u x ξ 1 t u t u x ξ 1 u ζ = D (η u t D (ξ 0 u xd (ξ 1 ζ 1 = η x + u x η u u t ξx 0 u t u x ξu 0 u x ξx 1 u 2 xξu 1 ζ j = D j (ζ u tx D j (ξ 0 u x x kd j(ξ k ζ 11 = D x (ζ 1 u tx D x (ξ 0 u xx D x (ξ 1 = η xx + 2u x eta xu + u xx η u + u 2 xη uu 2u tx ξx 0 u t ξxx 0 2u t u x ξxu 0 (u t u xx + 2u x u tx ξu 0 u t u 2 xξuu 0 2u xx ξx 1 u x ξxx 1 2u 2 xξxu 1 3u x u xx ξu 1 u 3 xξuu 1 where D x and D t are total dervatves wth respect to the varables x and t respectvely and are defned by D x = x + u x u + u xx + u xt +. u x u t D t = t + u t u + u xt + u tt +. [9] u x u t The determnng equaton s gven by {ζ 0 + 1 2 +A2 xu xx ξ 1 + 1 2 A2 x 2 ζ 11 +Bu x ξ 1 +Bxζ 1 Cη} ut= 1 2 A2 x 2 u xx Bxu x+cu = 0 (2.4 31
The substtutons of ζ 0 t, ζ 1 and ζ 11 n the determnng equaton yelds that η t + ( 1 2 A2 x 2 u xx Bxu x + Cu(η u ξ 0 t ( 1 2 A2 x 2 u xx Bxu x + Cu 2 ξ 0 u u x ξ 1 t ( 1 2 A2 x 2 u xx Bxu x + Cuu x ξ 1 u + A 2 xu xx ξ 1 + 1 2 A2 x 2 η xx + A 2 x 2 u x η xu + 1 2 A2 x 2 u xx (η u 2ξ 1 x 3u x ξ 1 u + 1 2 A2 x 2 u 2 x(η uu 2ξ 1 ux A 2 x 2 u tx ξ 0 x 1 2 A2 x 2 ( 1 2 A2 x 2 u xx Bxu x + Cuξ 0 xx A 2 x 2 ( 1 2 A2 x 2 u xx Bxu x + Cuξ 0 xu 1 2 A2 x 2 u xx ( 1 2 A2 x 2 u xx Bxu x + Cuξ 0 u 1 2 A2 x 2 ( 1 2 A2 x 2 u xx Bxu x + Cuξ 0 uu 1 2 A2 x 2 u 3 xξ 1 uu + Bv x ξ 1 A 2 x 2 u tx u x ξ 0 u + Bxu x (η u ξ 1 x Bx( 1 2 A2 x 2 u xx Bxu x + Cuξ 0 x + Bxη x Bxu x ( 1 2 A2 x 2 u xx Bxu x + Cuξ 0 u Bxu 2 xξ 1 u Cη = 0. (2.5 The soluton of determnng equaton (2.5 provdes an nfnte dmensonal vector space of nfnte symmetres of the equaton (2.1 spanned by the operators and X 1 = 2A 2 t 2 t + 2A2 tx ln x x + {(ln x Dt2 + 2A 2 Ct 2 A 2 t}u u X 2 = 2t + x(ln x + Dt t x + (Ctu u X 3 = t X 4 = A 2 xt + {ln x Dt}u x u X 5 = x x X 6 = u u, X = g(t, x u (2.6 (2.7 The functon g(t, x n (2.7 s an arbtrary soluton of equaton (2.1 and X s an nfnte symmetry [9]. 2.1.1 Fnte symmetry transformatons The fnte symmetry transformatons t = f(t, x, u, ɛ, x = g(t, x, u, ɛ, ū = h(t, x, u, ɛ, 32
correspondng to the generators (2.6 and (2.7 are found by solvng the Le equatons (1.8. These are gven by: and t X 1 : t = 1 2A 2 ɛ 1 t, x = x t 1 2A 2 ɛ 1t, ū = u 1 2A 2 ɛ 1 te [(ln x Dt2 +2A 2 Ct 2 ]ɛ 1 1 2A 2 ɛ 1 t X 2 : t = tɛ 2 2, x = xɛ 2 e D(ɛ2 2 ɛ2t, ū = ue C(ɛ2 2 ɛ 2t ɛ 2 0 X 3 : t = t + ɛ 3, x = x, ū = u X 4 : t = t, x = xe A2 ɛ 4 t, ū = ux ɛ 4 e ( 1 2 A2 ɛ 2 4 Dɛ 4t X 5 : t = t, x = xɛ 5, ū = u ɛ 5 0 X 6 : t = t, x = x, ū = uɛ 6 ɛ 6 0 X φ : t = t, x = x, ū = u + g(t, x The functons ɛ 1,..., ɛ 6 are the parameters of the one-parameter groups generated by X 1,...X 6, and g(t, x s an arbtrary soluton of (2.1. The operators X 1,...X 6 generate a sx-parameter group and X φ generates an nfnte group. 2.2 Transformaton to heat equaton In the paper [9] as well the text [17] Gazzov and Ibragmov nform that equaton (2.1 s reducble to heat equaton by Le equvalence transformaton v τ = v yy (2.8 τ = β(t, y = α(t, x, v = γ(t, xu, α x 0, β t 0. (2.9 Wth change of varables (2.9 the equaton (2.8 becomes [ 2γx u xx + γ + α xα t α ] [ xx u x α2 x γxx u t + β t α x β t γ + α xα t γ x α2 xγ t β t γ β t γ α ] xxγ x u = 0 (2.10 α x γ Equaton (2.10 compared to (2.1 wrtten n the form u xx + 2B A 2 x u x + 2 A 2 x 2 u t 2C A 2 x 2 u = 0 33
results n the system γ xx γ + α xα t γ x β t γ 2γ x γ Solvng for α n (2.11 results n that + α xα t β t α2 xγ t β t γ α xxγ x α x γ αx 2 = 2 (2.11 β t A 2 x 2 α xx α x = 2B A 2 x (2.12 = 2C A 2 x 2 (2.13 α(t, x = ϕ(t A ln x + ψ(t, β t = 1 2 ϕ(t2 (2.14 wth ϕ(t and ψ(t as arbtrary functons. The substtuton of (2.14 n (2.12 results n γ(t, x = ν(tx B A 2 1 2 + ψ t Aϕ + ψ t 2A 2 ϕ ln x (2.15 wth an arbtrary functon ν(t. The soluton of (2.13 results n two possbltes, ether and the functon ν(t or and the functon ν(t ϕ = 1 L Kt, ψ = M + N, K 0, L Kt ν t ν = M 2 K 2 2(L Kt 2 K 2(L Kt A2 8 + B 2 B2 2A 2 C ϕ = L, ψ = Mt + N, L 0, ν t ν = M 2 L 2 K 2(L Kt A2 8 + B 2 B2 2A 2 C wth arbtrary constants K,L,M and N. Ths results n two dfferent transformatons that assocate (2.1 and (2.8. The frst transformaton s gven by y = ln x A(L Kt + M L Kt + N, τ = 1 2K(L Kt + P, K 0 v = E L Kte [ MK 2 whle the second transformaton s y = L A 2(L Kt 1 2 ( B A A B 2 C]t Ct x B A 2 1 2 + MK A(L Kt + ln x + Mt + N, τ = L2 2 t + P, L 0 K ln x 2A 2 (L Kt u (2.16 v = Ee M2 2L 2 1 2 ( B A A B 2t x B A 2 1 2 + M AL u (2.17 34
The calculaton presents two dfferent transformatons that assocate (2.1 and (2.8 ([9]. 2.3 Invarant Solutons One of the mportant components of classfcaton of symmetry analyss s the constructon of nvarant solutons. The paper ([9] llustrates the calculaton of nvarant soluton by consderng a subgroup and the procedure s appled to each operator and the results are as follows: The one-parameter subgroup wth the generator provdes nvarants gven by The nvarant soluton s of the form X = X 1 + X 2 + X 3 = t + x x + u u. I 1 = t ln x, I 2 = u x. I 2 = φ(i 1 or u = xφ(z, z = t ln x. The substtuton of these nvarants n equaton (2.1 results n the equaton wth constant coeffcents gven by A 2 φ + (1 B A 2 φ + (B Cφ = 0 whch s solvable. Applyng the procedure to ndvdual operators the results are as follows: X 1 : u = 1 t e (ln x Dt2 ( 2A 2 t +Ctφ( ln x t, φ = 0 hence whence u = (K 1 ln x t 3 2 + K 2 e t (ln x Dt2 ( 2A 2 +Ct t X 2 : u = e Ct φ( ln x t D t, A 2 φ zφ = 0, z = ln x t D t φ(z = K 1 z 0 e µ2 2A 2 dµ + K 2 35
X 3 : u = φ(x, 1 2 A2 x 2 φ + Bxφ Cφ = 0. The equaton reduces to lnear coeffcents when z = ln x s ntroduced. X 4 : u = e (ln x Dt2 ( 2A 2 t φ(t, φ + ( 1 2t Cφ = 0 whence and hence u = K t e φ = K t e Ct (ln x Dt2 ( 2A 2 +Ct t X 5 : u = φ(t, φ Cφ = 0 (2.18 whence u = Ke Ct (2.19 K, K 1, K 2 are constants of ntegraton, and D = B 1 2 A2. Operators X 6, X do not provde nvarant solutons. 36
Chapter 3 Introducton of new method Ths chapter presents the Symmetry Analyss of transformed one-dmensonal and two-dmensonal Black-Scholes equatons. The method employs a formula that we shall henceforth be refer to as the Manale s formula. The ratonale and justfcaton of Manale s formula s presented n Appendx A. We determne the symmetres from a modfed transformaton wth an nfntesmal ω 0. We compute the Commutator Table for operators n one-dmensonal and determne nvarant solutons for one operator n each case. Graphcal solutons are presented for the calculated solutons. 3.1 The transformed one-dmensonal Black-Scholes equaton We present a new method to Symmetry Analyss of (2.1 where we start wth transformng the equaton n a dfferent ndependent varable, r. The one-dmensonal Black-Scholes equaton (2.1 s transformed usng the followng change of varables. u x = u x xu x = u x x = u ln x (3.1 37
Let We therefore express Also, Therefore r = ln x, r x = 1 x x r = x xu x = u r u xx = x ( u x x 2 u xx = x 2 ( u x x ( u } = x{ x x x { ( u } = x ln x x { ( u } = x r x { ( 1 u } = x r x ln x { = x 1 r u x 2 { = x 1 x = u r + 2 u r 2 then x r + 1 x u r + 1 2 u } x r 2 { ( 1 = x r x 2 u } r 2 u } r (3.2 (3.3 (3.4 xu x = u r x 2 u xx = u rr u r (3.5 where r s gven by equaton (3.2. We substtute equaton (3.5 n equaton (2.1 and defne D = B A2 2, then the Black-Scholes one dmensonal equaton transforms to u t + 1 2 A2 u rr + Du r Cu = 0. (3.6 38
3.2 Soluton of determnng equaton for (3.6 The nfntesmal generator for pont symmetry admtted by equaton (3.6 s of the form X = ξ 1 (t, r t + ξ2 (t, r r + η(t, r u The extenson of the equaton up to the second prolongaton s gven by (3.7 X (2 = X + η (1 t where X s defned by equaton (3.7. The determnng equaton s gven by + η (1 r + η (2 rr (3.8 u t u r u rr η (1 t + 1 2 A2 η (2 rr + Dη (1 r Cη = 0 (3.9 when u rr = ( 2 A 2 [u t + Du r Cu] (3.10 where we defne the followng from ([6],[15] η = fu + g η (1 t = g t + f t u + [f ξ 1 t ]u t ξ 2 t u r η (1 r = g r + f r u + [f ξ 2 r]u r ξ 1 ru t (3.11 η (2 rr = g rr + f rr u + [2f r ξ 2 rr]u r ξ 1 rru t + [f 2ξ 2 r]u rr 2ξ 1 ru tr The substtutons of η (1 t, η r (1 and η rr (2 n the determnng equaton yelds that g t + f t u + [f ξ 1 t ]u t ξ 2 t u r + ( 1 2 A2 {g rr + f rr u + [2f r ξ 2 rr]u r ξ 1 rru t + [f 2ξ 2 r]( 2 A 2 [u t + Du r Cu] 2ξ 1 ru tr } (3.12 + (D[g r + f r u + [f ξ 2 r]u r ξ 1 ru t ] Cfu Cg = 0 We set the coeffcents of u r, u tr, u t and those free of these varables to zero. We thus 39
have the followng monomals whch we termed defnng equatons u tr : ξr 1 = 0, (3.13 u t : ξt 1 + 2ξr 2 = 0 (3.14 u r : ξt 2 + A 2 f r + Dξr 2 1 2 A2 ξrr 2 = 0, (3.15 u 0 r : g t + 1 2 A2 g rr + Dg r Cg = 0, (3.16 u : f t + 1 2 A2 f rr + Df r 2Cξr 2 = 0 (3.17 From defnng equatons (3.13 and (3.14 we have that ξrr 2 = 0 (3.18 Thus ξ 2 = ar + b (3.19 whch can be expressed usng Manale s formula wth nfntesmal ω as ξ 2 = ωr ωr a sn( + bφ cos( ω, where φ = sn( ω, (3.20 and a and b are arbtrary functons of t. We dfferentate equaton (3.20 wth respect to r and t and obtan the followng equatons ξ 2 r = a cos( ωr ξ 2 rr = ω ξ 2 t = bφ sn(ωr, (3.21 a sn( ωr ω b φ cos(ωr, (3.22 ωr ωr ȧ sn( + ḃφ cos( ω and from defnng equaton (3.14 we have ξ 1 t = 2ξ 2 r whch mples that ξ 1 = 2at cos( ωr (3.23 2btφ sn(ωr + C. (3.24 We substtute equatons (3.21, (3.22 and (3.23 n the defnng equaton (3.15 to get the expresson for f r gven by f r = cos( ωr { ωbφ 2 ḃφ A 2 ω Da A 2 } + sn( ωr { ωa 2 ȧ A 2 ω + Dbφ } A 2 (3.25 40
Integratng equaton (3.25 wth respect to r gves the expresson for f f = sn( ωr { bφ 2 ḃφ A 2 ω Da } + cos( ωr { a 2 A 2 ω 2 + ȧ A 2 ω Dbφ } + k(t (3.26 2 A 2 ω We use equatons (3.25 and (3.26 to get expressons for f rr and f t gven by f rr = sn( ωr { ω2 bφ ḃφ 2 A + Daω } + cos( ωr { ω 2 2 A 2 a 2 + ȧ A + Dbωφ } 2 A 2 (3.27 and f t = sn( ωr { ḃφ 2 bφ A 2 ω Dȧ } + cos( ωr {ȧ 2 A 2 ω 2 + ä A 2 ω Dȧφ } + k (t (3.28 2 A 2 ω We substtute equatons (3.21, (3.25, (3.27 and (3.28 nto the defnng equaton (3.17 and solve the equaton sn( ωr { ḃφ 2 + k (t + sn( ωr { ba2 ω 2 φ 4 bφ A 2 ω 2 Dȧ A 2 ω + 2Cbφ } + cos( ωr {ȧ 2 + ä A 2 ω 2 Dḃφ ḃφ 2 + Daω 2 } A 2 ω 2Ca } + cos( ωr { aa 2 ω 2 + ȧ 4 2 + Dbωφ 2 cos( ωr { Dbωφ Dḃφ 2 A 2 ω D2 a } + sn( ωr { A 2 Daω Dȧ 2 A 2 ω + D2 bφ } = 0 A 2 (3.29 } + We collect all the coeffcents of sne functon together and equate them to zero. A smlar step s taken wth the cosne functon. For the coeffcents of sne functon we have: ḃφ 2 + Daω 2 bφ A 2 ω Dȧ 2 A 2 ω ba2 ω 2 φ ḃφ 4 2 Daω 2 Dȧ A 2 ω D2 bφ + 2Cbφ = 0 A 2 whch smplfes to a second-order ordnary lnear dfferental equaton (3.30 b + ḃa 2 ω 2 + ba4 ω 4 4 D2 b A 2 + 2CA2 ω 2 b = 0 (3.31 Solvng equaton (3.31 we proceed as follows. Let β = A2 ω 2 2, and k 1 = D2 2C (3.32 A2 We also set α 1 = bβ 2 k, then α 1 = ḃβ2, α 1 = bβ 2 (3.33 41
Equaton (3.31 transforms to α 1 + 2α 1 + α 1 β 2 = 0. (3.34 To fnd the soluton of equaton (3.34 we proceed as follows. We set α 1 = cz (3.35 where c = c(t, z = z(t. Then α 1 = c z + cz (3.36 α 1 = c z + 2c z + cz (3.37 We substtute equatons (3.35, (3.36 and (3.37 nto equaton (3.34 and after rearrangng we solve the equaton cz + (2c + 2c z + (c + 2c + β 2 cz = 0 (3.38 The choce for c s such that whence 2c + 2c = 0, (3.39 c = e t. (3.40 The equaton (3.38 smplfes to z + (β 2 1z = 0 (3.41 The soluton for equaton (3.34 s now wrtten ( α 1 = e t sn ω cos ωt C 1 ω t sn ωt + C 2 e ω (3.42 so that when β = ±1 or ω 0 the soluton for z s lnear, and we defne ω = β 2 1 (3.43 We substtute for b n equaton (3.33 to obtan that {( b = e t sn ω cos ωt t sn ωt } C β 2 1 + C 2 e + D2 ω ω β 2 A + 4C 2 β (3.44 42
Smlarly for the coeffcents of the cosne functon we have ȧ 2 + äφ A 2 ω Dḃφ 2 A 2 ω + aa2 ω 2 φ + ȧ 4 2 + Dbωφ Dbωφ 2 2 Dḃφ A 2 ω D2 a A + 2aC = 0 2 whch smplfes to a second-order ordnary lnear dfferental equaton (3.45 ä + ȧa 2 ω 2 + aa4 ω 4 4 ad2 A 2 2aA2 ω 2 C = 0 (3.46 Solvng equaton (3.46 we fnd the soluton for a to be {( a = e t sn ω cos ωt C β 2 3 ω sn ωt } + C 4 + D2 ω β 2 A 4C 2 β (3.47 and we also have that k (t = 0 k(t = C 5 (3.48 We dfferentate equatons (3.44 and (3.47 to obtan expressons for ȧ and ḃ ( ȧ = e t sn ω cos ωt C β 2 3 ω sn ωt ( + C 4 + e t C ω β 2 3 sn ω sn ωt + C 4 cos ωt (3.49 Smlarly ( ḃ = e t sn ω cos ωt C β 2 1 ω sn ωt ( + C 2 + e t C ω β 2 1 sn ω sn ωt + C 2 cos ωt (3.50 We substtute equatons (3.32,(3.44,(3.47,(3.48,(3.49 and (3.50 nto equaton (3.26 and get the expresson for f gven as f = sn( ωr { C 1φe t sn ω cos ωt 2β 2 ω C 2φe t sn ωt 2β 2 ω D2 φ 2β 3 A 4Cφ 2 β + C 1φe t sn ω cos ωt 2β 3 ω C 2φe t cos ωt 2β 3 + C 2φe t sn ωt 2β 3 ω C 3Dωe t sn ω cos ωt 2β 3 + C 1φe t sn ω cos ωt 2β 3 C 4Dωe t sn ωt 2β 3 + cos( ωr { C3 e t sn ω cos ωt 2β 2 ω + C 4e t sn ωt 2β 2 ω + D2 2β 3 A 4Cφ 2 β C 3e t sn ω cos ωt 2β 3 ω C 4e t cos ωt 2β 3 C 2φDωe t sn ωt 2β 3 C 4e t sn ωt 2β 3 ω C 1φDωe t sn ω cos ωt 2β 3 D3 φω } + C 2β 3 A 2 5 C 3e t sn ω sn ωt 2β 3 D3 ω 2β 3 A 2 } 43
3.2.1 Infntesmals for equaton (3.6 The lnearly ndependent solutons of the defnng equatons (3.12 lead to the nfntesmals ( ωr { 2te ξ 1 t = cos β 2 ω ( ωr { 2tφe t sn β 2 ω } + 2tφD2 8Ct β 2 A 2 β {( } C 3 sn ω cos ωt + C 4 sn ωt + 2tD2 {( C 1 sn ω cos ωt cos ( ωr ( ωr { e ξ 2 t = sn β 2 ω ω ( ωr { e t φ + cos β 2 ω ω + 4C ( ωr ωβ sn + 4Cφ ωβ + 8Ctφ β sn } + C 2 sn ωt ( ωr + C 6 β 2 A 2 } ( } C 3 sn ω cos ωt + C 4 sn ωt D2 ωβ 2 A 2 ( C 1 sn ω cos ωt C 2 sn ωt ( ωr cos D2 φ } ωβ 2 A 2 f = sn( ωr { C 1φe t sn ω cos ωt 2β 2 ω C 2φe t sn ωt 2β 2 ω D2 φ 2β 3 A 4Cφ 2 β + C 1φe t sn ω cos ωt 2β 3 ω C 2φe t cos ωt 2β 3 + C 2φe t sn ωt 2β 3 ω C 3Dωe t sn ω cos ωt 2β 3 + C 1φe t sn ω cos ωt 2β 3 C 4Dωe t sn ωt 2β 3 + cos( ωr { C3 e t sn ω cos ωt 2β 2 ω + C 4e t sn ωt 2β 2 ω + D2 2β 3 A 4Cφ 2 β C 3e t sn ω cos ωt 2β 3 ω C 4e t cos ωt 2β 3 C 2φDωe t sn ωt 2β 3 C 4e t sn ωt 2β 3 ω C 1φDωe t sn ω cos ωt 2β 3 D3 φω } + C 2β 3 A 2 5 C 3e t sn ω sn ωt 2β 3 D3 ω 2β 3 A 2 } (3.51 (3.52 (3.53 44
3.2.2 The symmetres for equaton (3.6 Accordng to (3.12, the nfntesmals: (3.53, (3.51 and (3.52, lead to the generators ( X 1 = { + + e t φ 2β 3 2te t φ β 2 ω ( ωr sn ω cos ωt sn e t φ ( ωr 2β 2 ω sn ω cos ωt sn sn ω cos ωt sn ( ωr ( e t t + φ ( ωr β 2 ωω sn ω cos ωt cos + e t φ ( ωr 2β 3 ω sn ω cos ωt sn Dφωe t sn ω cos ωt 2β 3 cos ( ωr } u u r (3.54 ( X 2 = { + 2te t φ β 2 ω ( ωr sn ωt sn e t φ ( ωr 2β 2 ω sn ωt sn ( e t t φ ( ωr β 2 ωω sn ωt cos + e t φ ( ωr 2β 3 ω sn ωt sn r e t φ ( ωr 2β cos ωt sn Dφωe t sn ωt ( ωr } cos u 3 2β 3 u ( 2te t ( ωr ( e t X 3 = β 2 ω sn ω cos ωt cos t + ( ωr β 2 ωω sn ω cos ωt cos r { e t ( ωr ( + 2β 2 ω sn ω cos ωt cos e t ωr 2β 3 ω sn ω cos ωt cos ( e t ωr 2β sn ω sn ωt cos Dωe t sn ω cos ωt ( ωr } sn u 3 2β 3 u ( 2te t φ ( ωr ( e t X 4 = β 2 ω sn ωt cos t + φ ( ωr β 2 ωω sn ωt sn r { e t ( ωr ( + 2β 2 ω sn ωt cos e t ωr 2β 3 ω sn ωt cos ( + e t ωr 2β cos ωt cos Dωe t sn ωt ( ωr } sn u 3 2β 3 u ( ( X 5 = 2tD2 ωr ( ωr ( ( cos φ sn β 2 A 2 t + D2 ωr φ cos ωβ 2 A 2 ( ωr + sn r D2 ω ( ( ωr ( ωr sn + φ cos 2β 3 A 2 ( ωr ( ωr + D sn + Dφ cos u u X 6 = u u (3.55 (3.56 (3.57 (3.58 (3.59 45
X 8 = 8Ct β ( φ sn + 4C β ( cos X 7 = t ( ωr ( ωr + φ sn ( ωr u u ( ωr cos t + 4Ct βω ( φ cos The defnng equaton (3.16 gves an nfnte symmetry X = g(t, r u ( ωr + sn ( ωr r (3.60 (3.61 (3.62 3.2.3 Table of Commutators The determned set of operators (3.54 to (3.61 form a Le Algebra f ther commutator s blnear, ant symmetrc and satsfy the Jacob dentty as stated n (1.6. The commutator for the par of operators (3.54 and (3.55 s determned as follows: [X 1, X 2 ] = { ( 2te t φ X 1 β 2 ω sn ωt sn ( ωr ( 2te t φ X2 β 2 ω sn ω cos ωt sn ( ωr } t + { ( e t φ X 1 β 2 ωω sn ωt cos ( ωr ( e t φ X2 β 2 ωω sn ω cos ωt cos ( ωr } r + { ( e t φ X 1 2β 2 ω sn ωt sn ( ωr e t φ + 2β 3 ω sn ωt sn ( ωr + e t φ 2β cos ωt sn ( ωr Dφωe t sn ωt cos ( ωr 3 2β 3 ( e t φ X 2 2β 2 ω sn ω cos ωt sn ( ωr e t φ + 2β 3 ω sn ω cos ωt sn ( ωr + e t φ 2β sn ω cos ωt sn ( ωr Dφωe t sn ω cos ωt cos ( ωr } u (3.63 3 2β 3 u The equaton (3.63 smplfes to ( 4t 2 e 2t [X 1, X 2 ] = β 4 ω sn2 ω sn ( 2 ωr t + 0 r + 0 u u = 0 as ω 0. Thus we have that [X 1, X 2 ] = 0 (3.64 Smlarly we determne the other pars of commutators that are defned and end wth the Commutator Table gven as n Fgure (3.1. The Commutator Table s determned for that nstant ω 0. 46