New symmetries of Black-Scholes equation

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1 Proceedngs of he 03 Inernaonal Conference on Appled Mahemacs and Compuaonal Mehods New symmeres of Black-Scholes equaon TSHIDISO MASEBE Tshwane Unversy of Technology Mahs,Scence& Tech Deparmen No Aubrey Malala Road, Soshanguve H Souh Afrca masebep@u.ac.za JACOB MANALE Unversy of Souh Afrca Deparmen of Mahemacal Scences Preller sree 3, Cy of Tshwane Souh Afrca manaljm@unsa.ac.za Absrac: Le Pon symmeres and Euler s fmula f solvng second der dnary lnear dfferenal equaons are used o deermne symmeres f he one-dmensonal Black- Scholes equaon. One symmery s ulzed o deermne an nvaran soluons Key Wds: Le Pon Symmeres, Black- Scholes equaon, nvaran soluon. Inroducon The pas few years he wld experenced an economc meldown n par due o napproprae managemen of fnancal secures. A dervave fnancal secury may be defned as a secury whose value depends on he value of oher me basc underlyng varables whch may be prced raded secures, prces of commodes sock ndces [5]. The Black-Scholes equaon s a paral dfferenal equaon ha governs he value of fnancal dervaves. Deermnng he value of dervaves had been a problem n fnance f almos 70 years snce 990. In he early 70s, Black and Scholes made a poneerng conrbuon o fnance by developng a Black- Scholes equaon under very resrcve assumpons and he opon valuaon fmula. Scholes obaned a Nobel Prze f economcs n 997 f hs conrbuon Black had passed on n 995 and could no receve he prze personally [5].The wdely used one-dmensonal model one sae varable plus me s descrbed by he equaon u A x u xx Bxu x Cu = 0 wh consan coeffcens A, B and C. [] of he one-dmensonal Black-Scholes equaon and consruced nvaran soluons f some examples. In he presen projec we deermne he same usng Euler fmulas. The one-dmensonal Black-Scholes equaon s ransfmed usng he followng change of varables. Le We herefe express u x = u x xu x = u = u x ln x x r = ln x, hen r x = x x r = x 3 Le group hey s appled n he mahemacal model of fnance. In her wk [], Ibragmov and Gazzov deermned he complee symmery analyss xu x = u r

2 Proceedngs of he 03 Inernaonal Conference on Appled Mahemacs and Compuaonal Mehods Also, u xx = x u x x u xx = x u x x u = x{ x x x { u = x ln x x { u = x r x { u = x r x ln x { = x r u x { = x x = u r u r Therefe x r x u r u x r { = x r x u r u r 5 xu x = u r x u xx = u rr u r 6 where r s gven by 3. We subsue f 8 n equaon and defne D = B A, hen he Black-Scholes one dmensonal equaon ransfms o Therefe u A u rr Du r Cu = 0. 7 xu x = u r x u xx = u rr u r 8 where r s gven by 3. We subsue f 8 n equaon and defne D = B A, hen he Black-Scholes one dmensonal equaon ransfms o u A u rr Du r Cu = 0. 9 Soluon of deermnng equaon The nfnesmal genera f pon symmery admed by equaon 0 s of he fm X = ξ, r ξ, r r η, r u Is frs and second prolongaons are gven by X = X η 0 η r η rr u u r u rr where X s defned by equaon 0. The deermnng equaon s gven by when η A η rr Dη r Cη = 0 u rr = A [u Du r Cu] 3 where we defne he followng from [],[3] η = fu g η η r η rr = g f u [f ξ ]u ξ u r = g r f r u [f ξ r ]u r ξ r u = g rr f rr u [f r ξ rr]u r ξ rru [f ξ r ]u rr ξ r u r The subsuons of η, η r and η rr n he deermnng equaon yelds ha g f u [f ξ ]u ξ u r A {g rr f rr u [f r ξ rr]u r ξ rru [f ξ r ] A [u Du r Cu] ξ r u r D[g r f r u [f ξ r ]u r ξ r u ] Cfu Cg = 0 5 We se he coeffcens of u r, u r, u and hose free of hese varables o zero. We hus have he followng defnng equaons u r : ξr = 0, 6 u : ξ ξr = 0 7 u r : ξ A f r Dξ r A ξ rr = 0, 8 u 0 r : g A g rr Dg r Cg = 0, 9 u : f A f rr Df r Cξ r = 0 0

3 Proceedngs of he 03 Inernaonal Conference on Appled Mahemacs and Compuaonal Mehods Thus From defnng equaon 7 we have ha ξ rr = 0 ξ = ar b whch can be expressed usng Euler fmula wh nfnesmal ω as ξ a sn bϕ cos = ω where ϕ = sn ω 3. We dfferenae equaon 3 wh respec o r and and oban he followng equaons ξ rr = ω ξ r = a cos ξ = bϕ sn, a sn ω b ϕ cos, 5 ȧ sn ḃϕ cos ω and from defnng equaon 7 we have ξ whch mples ha ξ = a cos 6 = ξ r bϕ sn C. 7 We subsue f equaons, 5 and 6 n he defnng equaon 8 o ge he expresson f f r gven by f r = cos { ωbϕ sn { ωa ḃϕ A ω Da A ȧ A ω Dbϕ 8 A Inegrang equaon 8 wh respec o r gves he expresson f f f = sn { bϕ cos { a ḃϕ A ω Da A ω ȧ A ω Dbϕ A ω k 9 We use he equaons 8 and 9 o ge he expressons f f rr and f gven by f rr = sn { ω bϕ ḃϕ A Daω A cos { ω a ȧ A Dbωϕ 30 A and f = sn { ḃϕ bϕ A ω Dȧ A ω cos {ȧ ä A ω Dȧϕ A ω k 3 We subsue f he equaons, 8, 30 and 3 n he defnng equaon 0 and solve he equaon sn { ḃϕ bϕ A ω Dȧ A ω Cbϕ cos {ȧ ä A ω Dḃϕ A ω Ca k sn { ba ω ϕ ḃϕ Daω cos { aa ω ȧ Dbωϕ cos { Dbωϕ Dḃϕ A ω D a A sn { Daω Dȧ A ω D bϕ A = 0 3 We collec all he coeffcens of sne funcon ogeher and equae hem o zero. Smlarly wh he cosne funcon. F he coeffcens of sne funcon we have: ḃϕ Daω bϕ A ω Dȧ A ω ba ω ϕ ḃϕ Daω Dȧ A ω D bϕ A Cbϕ = 0 33 whch smplfes o a second-der dnary lnear dfferenal equaon b ḃa ω ba ω D b A CA ω b = 0 3 Solvng f equaon 3 we proceed as follows. Le We also se β = A ω, and k = D A C α = bβ k, hen 35 α = ḃβ, 36 α = bβ 3

4 Proceedngs of he 03 Inernaonal Conference on Appled Mahemacs and Compuaonal Mehods Equaon 3 ransfms o α α α β = To fnd he soluon of equaon 37 we proceed as follows. We se α = cz 38 where c = c, z = z. Then α = c z cz 39 α = c z c z cz 0 whch smplfes o a second-der dnary lnear dfferenal equaon ä ȧa ω aa ω ad A aa ω C = 0 9 Solvng f equaon9 we fnd he soluon f a o be We subsue f equaons 38, 39 and 0 n equaon 37 and afer rearrangng we solve he equaon cz c c z c c β cz = 0 The choce f c s such ha a = e { sn cos β C 3 D β A C β sn C 50 whence The equaon smplfes o c c = 0, c = e. 3 z β z = 0 The soluon f 37 s now wren α = e sn cos C sn C e 5 so ha when β = ± ω 0 he soluon f z s lnear, and we defne = β 6 We subsue f b n equaon 36 o oban ha b = e { sn cos β C sn C e D β A C β 7 Smlarly f he coeffcens of he cosne funcon we have ȧ äϕ A ω Dḃϕ A ω aa ω ϕ ȧ Dbωϕ Dbωϕ Dḃϕ A ω D a A ac = 0 8 and we also have ha k = 0 k = C 5 5 We dfferenae equaons 7 and 50 o oban expressons f ȧ and ḃ ȧ = e sn cos β C 3 C sn Smlarly e β C 3 sn sn C cos 5 ḃ = e sn cos sn β C C e 53 β C sn sn C cos We subsue f equaons 35,7,50,5,5 and 53 n equaon 9 and ge he expresson f

5 Proceedngs of he 03 Inernaonal Conference on Appled Mahemacs and Compuaonal Mehods f gven as f = sn { C ϕe sn cos β C ϕe sn β C ϕe sn C ϕe cos C Dωe sn D ϕ A Cϕ β C ϕe sn cos C ϕe sn cos C 3Dωe sn cos D3 ω A cos { C3 e sn cos β C e sn β D A Cϕ β C 3e sn cos C e sn C 3e sn sn C e cos C ϕdωe sn C ϕdωe sn cos D3 ϕω A C 5 5 f = sn { C ϕe sn cos β C ϕe sn β Cϕ β D ϕ A C ϕe sn cos C ϕe sn C ϕe cos C Dωe sn C ϕe sn cos C 3Dωe sn cos D3 ω A cos { C3 e sn cos β C e sn β D A Cϕ β C 3e sn cos C e sn C e cos C ϕdωe sn C 3e sn sn C ϕdωe sn cos D3 ϕω A C Infnesmals The lnearly ndependen soluons of he defnng equaons 5 lead o he nfnesmals { e ξ { = cos β C 3 sn cos C sn D β A { ϕe sn β { C sn cos C sn ϕd β A C 6 { e ξ = sn β ω C sn D ωβ A { e ϕ cos β ω C sn D ϕ ωβ A C 3 sn cos C sn cos The symmeres Accdng o 5, he nfnesmals: 57, 55 and 56, lead o he generas X = e ϕ β sn cos sn e ϕ β ω sn cos cos r { e ϕ β sn cos sn e ϕ sn cos sn e ϕ sn cos sn Dϕωe sn cos cos u u 58 5

6 Proceedngs of he 03 Inernaonal Conference on Appled Mahemacs and Compuaonal Mehods X = e ϕ β sn sn e ϕ β ω sn cos r { e ϕ β sn sn e ϕ sn sn e ϕ cos sn Dϕωe sn cos u u e X 3 = β sn cos cos e β ω sn cos cos r { e β sn cos cos e sn cos cos e ϕ sn sn cos Dωe sn cos sn u u e ϕ X = β sn cos e ϕ β ω sn sn r { e β sn cos e sn cos e cos cos Dωe sn sn u u X 5 = D β A cos D ωβ A ϕ cos D ω A sn ϕ sn sn ϕ cos X 6 = u u r u u X 8 = C β ϕ sn X 7 = cos u u 6 65 The defnng equaon 9 gves an nfne symmery X = g, r u 66.3 Invaran soluons hrough he symmery X 3 The nvarans are deermned from solvng he equaon e I X 3 I = β sn cos cos e I β ω sn cos cos r { e β sn cos cos e sn cos cos e ϕ sn sn cos Dωe sn cos sn u I u = 0 The characersc equaon of 67 s gven by d e sn cos cos = β dr e sn sn cos β ω = du ue { β sn cos cos sn cos cos sn sn cos Dω sn cos sn

7 Proceedngs of he 03 Inernaonal Conference on Appled Mahemacs and Compuaonal Mehods From equaon 68 we have ha smplfes o d e sn cos cos = whose soluon s β dr e sn sn cos β ω The frs nvaran s gven by 69 d = ω dr 70 = Ce 7 ψ = e From equaon 68 we also have ha dr e sn sn cos β ω = du ue { β sn cos cos sn cos cos sn sn cos Dω sn cos sn Equaon 73 smplfes o β Dω an ω ω β dr = du u We negrae equaon 7 and oban We approxmae β β β β Dω ln cos C = ln u β β β β β Dω ln cos β Inegrang equaon 73 we oban u e The equaon 77 smplfes o u e = C 77 = ψ 78 whch s our second nvaran. If we defne ψ = hψ 79 where ψ s gven by equaon 7, hen an nvaran soluon s gven by u = e hψ 80 We dfferenae equaon 80 wh respec o and wce wh respec o r and ge he followng expressons f u, u r and u rr. u = e 3 h ψ 8 u r = ω e hψ ω e 3 h ψ 8 u rr = ω e hψ ω e 3 h ψ 83 6ω e 3 h ψ ω e 5 h ψ We subsue f equaons 8, 8 and 83 n he gnal equaon 9 and ge he followng equaon e 3 h ψ ω e hψ A ω e 3 A ω e 5 h ψ Dω e h ψ 3A ω hψ Dω e 3 h ψ e 3 h ψ Ce hψ = 0 8 Equaon 8 s a second der equaon n hψ. We rearrange n he der of dervaves of hψ, and apply equaon 35 β e 5 h ψ h ψ { β e 3 Dω βe e 3 6β e 3 hψ { Dkω Ce = 0 e e

8 Proceedngs of he 03 Inernaonal Conference on Appled Mahemacs and Compuaonal Mehods Leng ω 0 equaon 85 smplfes o β h ψ β h ψ 6β h ψ βhψ Chψ = 0 86 whch can be smplfed o βh ψ 0βh ψ β Chψ = Fgure : Plo of he soluon n 0. We elmnae from equaon 87 by applyng he followng change of varables. From equaon 7 we le hen and dλ = dψ, hψ = h 88 h ψ = h λ 89 h ψ = = d dψ h λ dh λ dψ = d dψ h λ dh λ dλ = {h λλ h λ e F ω 0 equaon 90 smplfes o d dψ {h λ 90 h ψ = {h λλ h λ 9 The subsuon of equaons 89, 9 ransfms equaon 87 o βh λλ 6βh λ β Ch = 0. 9 The soluon o equaon 9 s gven by h = C e 3β C e 3β 9β ββc λ β 9β ββc λ β Thus he nvaran soluon s u = e {C e 3β 9β ββc λ β C e 3β 9β ββc β λ However he equaon 9 can be expressed as u = e {C e 3β 9β ββc λ β C e 3β 9β ββc β λ Ths smplfes o u = e where = {C e 3 λ e λ C e 3 9β ββ C β λ e λ Snce < 0, we express equaon 96 as u = e {C e 3 λ sn λ C e 3 λ cos λ We however advance he same reason ha f equaon 97 o reurn o he lnear fm as 0 has o be ransfmed o be u = e {C e 3 λ sn λ C e 3 λ ϕ cos λ where ϕ = sn Soluons f equaon 98 Ths equaon 98 has some few soluons as ω 0. We recall ha. Soluon λ = ψ = e ψ ψ ψ = C 0 e d e 99 d ln u = Ae e 3 λ sn λ 00 = Ae e 3 C 0e ln sn C 0 e as ω 0, he soluon becomes ln u = 3 C 0 0 One of he assumpons of he Black-Scholes model s ha he opon value s perfecly lnear. The 8

9 3 3 Proceedngs of he 03 Inernaonal Conference on Appled Mahemacs and Compuaonal Mehods 8 6 Fgure : Plo of he soluon n 03. Fgure 3: Plo of he soluon n 08. lneary of he graph llusraes an mpan feaure of he Black-Scholes model n ha provdes an excellen approxmaon o he value of he opon wh varable volaly as long as mahemacal expecaon of he volaly s known []. Soluon u = Ae e 3 λ sn λ 0 = Acos sne 3 λ sn λ as ω 0, he soluon becomes u = Ae 3 λ sn λ 03 Ths nvaran soluon s conssen wh one of he soluons obaned by Ibragmov and Gazzov n her paper []. The plo of hs nvaran soluon s gven n Fgure..3 Soluon 3 u = Aωe e 3 ψ ω ψ dψ sn ψ ω ψ dψ ω = Aωe e 3 ψ ω ψ dψ d dω [sn = Aωe e 3 ψ ω ψ dψ dψ dω = Aωe e 3 = Aωe e 3 = Aω Bu ωe 3 ω e 3 3 re ψ ω ψ dψ re ψ ω ψ dψ re ψ ω d dω [sn = ω cos3 ω sn3 = ω cos3 ω sn3 = ω {ω3 cos3 ω 3 sn3 ψ dψ ] ψ ω ψ dψ ] [ cos ψ ω ψ dψ ] 0 05 and ω 3 r cos = ω e 3 ω e 3 = ω r ω e 3 06 e 3 = ω r e3 We subsue n equaon 0 and oban u = Aω r ω r e3 07 as ω 0, he soluon becomes u = A r 08 The plo of hs nvaran soluon s gven n Fgure 3.. Concluson In hs paper, new symmeres were obaned f he Black-Scholes equaon, and one was used o deermne group nvaran soluons. Some of he symmeres are comparable o he ones []. 3 APPENDIX A: Euler s fmulas and he nfnesmal ω I s well-known ha Le s group heecal mehods seek o reduce procedures f solvng dfferenal equaons of any challengng fm o smple ones ha may also have he fm a 0 ÿ b 0 ẏ c 0 y = 0, 09 f y = yx, wh parameers a 0, b 0 and c 0. I s also ha acceped Euler s fmulas are suable f solvng such equaons. They are: y = e b 0 x a 0 Ae ωx Be ωx, b 0 > a 0c 0, A Bx, b 0 = a 0c 0, e b 0 a 0 x [A cos ωx] Be b 0 a 0 x [sn ωx], b 0 < a 0 c 0 0 9

10 Proceedngs of he 03 Inernaonal Conference on Appled Mahemacs and Compuaonal Mehods where ω = b 0 a 0c 0 /a 0. Bu here s a problem wh hs sysem: I does no reduce o y = A Bx when b 0 = c 0 = 0. Ths s because Euler dd no solve he equaon o ge he fmulas. There has never been a need o do so, prmarly because he fmulas have been very successful n applcaons, and hey sll are. The need f an exac soluon here, s drven by he desre undersand soluons f equaon 9 hrough symmery mehods. I s mpossble hrough Euler s fmulas. To ge such exac fmula, frs le y = βz, wh β = βx and z = zx, so ha and ẏ = βz βż, ÿ = βz βż β z. These ransfm 09 no a 0 βz βż β z b 0 βz βż c 0 βz = 0. Tha s, a 0 β z a 0 β b0 β ż a 0 β b0 β c0 β z = 0. Choosng β o sasfy a 0 βb0 β = 0 smplfes equaon. Tha s, β = C 00 e b 0 a 0 x, f some consan C 00. Equaon assumes he fm z = a 0 β b 0 β c0 β z. a 0 β Tha s, z = b 0 a 0 c 0 z. Bu z can be wren as żdz/dx. Therefe, Tha s, ż dż dz = żdż = ż = b 0 a 0 c 0 b 0 a 0 c 0 z, zdz. b 0 a 0 c 0 z C 0, f some consan C 0. Tha s, b ż = 0 a 0 c 0 z C 0, Tha s, b 0 a0c0 dz = dx. z C 0 dz A 00 z = b 0 a 0c 0 wh A 00 = C 0/ b 0 a0c0. Hence, z = C 0 b 0 a0c0 sn f some consan C 0. Tha s, y = C 00 e b 0 x C a 0 0 b 0 a0c0 Leng = b 0 a 0c 0 sn b 0 a 0c 0 we have y = C 00 e b 0 x a 0 C 0 sn x C 0, y = C 00 e b 0 x a 0 sn x ]. C 0 dx, x C 0 b 0 a 0c 0 [ snc0 cos x cos C 0 A reducon o he rval case ÿ = 0 requres ha snc 0 = C 03 sn and cosc 0 = C 0 cos. Tha s, C03 C 0 =. Hence, y = C 00 e b 0 x a 0 C 0 cos smply sn x ], C 0 C03 sn [ cos x, x C 0 y = C 00 e b 0 a 0 x C0 C 03 sn cos x C 00 e b 0 a 0 x C0 C 0 sn x. I s very val o ndcae ha f he parameers n he denomna and sn are absbed no he coeffcens C 0 and C 03, hen fmula would reduce o one of Euler s fmulas. Bu he consequences would be faal, as fmula would no reduce o y = A Bx when b 0 = c 0 = 0, ha s, when = 0. Unfunaely, hs resul canno be found n any unversy exbook.. 30

11 Proceedngs of he 03 Inernaonal Conference on Appled Mahemacs and Compuaonal Mehods APPENDIX B: Useful lm resuls I s rue ha lm µ 0 { sn µ µ =. Ths can be wren n he fm { sn µx lm = 0, µ 0 µ lm µ 0 { sn µ µ µ cos = 0. Removng he lm f greaer clary: sn µ = µ µ cos. Tha s, We hen have sn µ = µ µ cos, 3 cos µ = sn µ. µ cos µ µ q = µ cos µ µ q. Carryng ou he dervave on he rgh hand sde: cos µ µ q = µ sn µ cos µ µ q. Subsung : cos µ µ q = µ Tha s, µ µ cos = µ cos cos µ cos µ µ q. µ cos µ, whch can be expressed n he fm µ µ µ cos µ 3 cos = µ sn. 5 Snce sn µ = 0 f µ small, follows hen ha µ cos µ = µ 3 cos µ. 6 Snce e µ ca be expressed n he fm cosµ/ snµ/, hen µ µ e µ = µ 3 cos, 7 so ha µe µ/ = µe µ/ = [ ] µ µ 3 cos, 8 [ ] µ µ 3 cos, 9 Therefe 9 and 3 can hen be wren n he fm µ u = [ µ 3 cos µ] ϕη, 0 wh µ = ω ω n he case of 9 and µ = ω ω f 3. Tha s, u = f 9, and f. References: u = ϕη ω ω ϕη ω ω [] Blumen,G.W. Anco,S.C. 00. Symmeres and Inegraon mehods f Dfferenal Equaons. New Yk. Sprnger-Verlag. [] Gazzov, R.K and Ibragmov, N.H.998. Le Symmery Analyss of Dfferenal Equaons n Fnance. Nonlnear Dynamcs 7: June. [3] Ibragmov, N.H.999. Elemenary Le Group Analyss and Ordnary Dfferenal Equaons. London. J. Wley & Sons Ld. [] Mller, R.M Opon Valuaon. Economc and Fnancal Modellng wh Mahemaca. Sprnger Verlag. [5] Slberberg,G.00.Dervave Prcng wh Symmery Analyss.hp:// Slberberg. pdf 3