An Approximation Formula for Basket Option Prices under Local Stochastic Volatility with Jumps: an Application to Commodity Markets

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1 CIRJE-F-973 An Approximaion Formula for Baske Opion Prices under Local Sochasic Volailiy wih Jumps: an Applicaion o Commodiy Markes Kenichiro Shiraya he Universiy of okyo Akihiko akahashi he Universiy of okyo May 215; Revised in June, July, Augus 215, and June 216 CIRJE Discussion Papers can be downloaded wihou charge from: hp:// Discussion Papers are a series of manuscrips in heir draf form. hey are no inended for circulaion or disribuion excep as indicaed by he auhor. For ha reason Discussion Papers may no be reproduced or disribued wihou he wrien consen of he auhor.

2 An approximaion formula for baske opion prices under local sochasic volailiy wih jumps: an applicaion o commodiy markes Kenichiro Shiraya, Akihiko akahashi June 2, 216 Absrac his paper develops a new approximaion formula for pricing baske opions in a local-sochasic volailiy model wih jumps. In paricular, he model admis local volailiy funcions and jump componens in no only he underlying asse price processes, bu also he volailiy processes. o he bes of our knowledge, he proposed formula is he firs one which achieves an analyical approximaion for he baske opion prices under his ype of he models. Moreover, in numerical experimens, we provide approximae prices for baske opions on he WI fuures and Bren fuures based on he parameers hrough calibraion o he plain-vanilla opion prices, and confirm he validiy of our approximaion formula. 1 Inroducion he baske opions are one of he mos popular exoic-ype opions in he commodiy and equiy markes. However, i is a ough ask o calculae a baske opion price wih compuaional speed fas enough for pracical purpose, mainly due o he difficuly of he analyical racabiliy and is high dimensionaliy. For insance, alhough he Mone Carlo mehod is easy o implemen, i requires a subsanial compuaional ime o obain an accurae value. Also, he numerical mehods for he parial differenial equaions PDEs have been well developed, bu i is sill very difficul o solve high dimensional PDEs wih accuracy and compuaional speed saisfacory enough in he financial business. o overcome he difficulies, his paper develops a new analyical approximaion formula for baske opions. In paricular, o he bes of our knowledge, our approximaion formula is he firs one which achieves a closed form approximaion of baske opions under sochasic volailiy models wih local volailiy funcions his research is suppored by JSPS KAKENHI Gran Number Graduae School of Economics, he Universiy of okyo Graduae School of Economics, he Universiy of okyo

3 and jump componens for no only he underlying asse price processes, bu also he volailiy processes. here exis a large number of preceding sudies on pricing baske opions. In he Black-Scholes model, Brigo, Mercurio, Rapisarda and Scoi 24 applied a momen maching mehod o approximae baske opion prices. Deelsra, Liinev and Vanmaele 24 derived he lower and upper bounds for baske call opion prices wih comonoonic approach. Under a Local volailiy model, akahashi 1999 showed an approximaion for baske opion prices wih an asympoic expansion echnique. Bayer and Laurence 214 used a hea kernel expansion and he Laplace approximaion mehod o derive very accurae approximae prices of baske opions. In a local volailiy jump-diffusion model, Xu and Zheng 21 derived a forward parial inegral differenial equaion PIDE for baske opion pricing and approximaed is soluion. Also, Xu and Zheng 213 applied he lower bound echnique in Rogers and Shi 1995 and he asympoic expansion mehod in Kuniomo and akahashi 21 o obain he approximae value of he lower bound of European baske call prices. Moreover, when he local volailiy funcion is ime independen, hey suggesed o have a closed-form expression for heir approximaion. Under a local sochasic volailiy model, Shiraya and akahashi 214 has developed a general pricing mehod for muli-asse cross currency opions which include cross currency opions, cross currency baske opions and cross currency average opions. hey also demonsraed ha he scheme is able o evaluae opions wih high dimensional sae variables such as 2 dimensions, which is necessary for pricing baske opions wih 1 underlying asses under sochasic volailiy environmen. Moreover, in pracice, fas calibraion is necessary in he opion markes relevan for he underlying asses and he currency, which was also achieved in he work. Models wihin he class of he so called local sochasic volailiy LSV model are mainly used in pracice: for example, SABR Hagan, Kumar, Lesniewskie, and Woodward 22, ZABR Andreasen and Huge 211, CEV Heson e.g. Shiraya e al. 212 and Quadraic Heson models e.g. Shiraya e al. 212 are well known. Noneheless, he LSV model is no always enough o fi o a volailiy smile and erm srucure. Hence, some advanced researches invesigaed a local sochasic volailiy wih jump model. Among hem, Eraker 24 found ha he models wih jump componens in he underlying price and volailiy processes showed beer performance in fiing o opion prices and he underlying price reurns daa simulaneously in sock markes. Pagliarani and Pascucci 213 derived an analyical approximaion of plainvanilla opion prices by applying he adjoin expansion mehod. However, o he bes of our knowledge no works have derived an analyical approximaion formula for he opion prices under a model which admis a local volailiy funcion and jumps boh in he underlying asse price and is volailiy processes. his paper develops a formula for pricing baske opions under he seing by exending an asympoic expansion approach. his closed form equaion has an advanage in making use of he beer calibraion o he raded individual opions whose underlying asses are included in a baske opion s underlying. 2

4 In fac, our numerical experimens provide esimaes of baske opion prices based on he parameers obained by calibraion o he marke prices of WI fuures opions and Bren fuures opions. hen, hose esimaed prices are compared wih he prices calculaed by Mone Carlo simulaions. An asympoic expansion approach in finance was iniiaed by Kuniomo and akahashi 1992, Yoshida 1992, and akahashi 1995,1999, which provides us a unified mehodology for evaluaion of prices and Greeks in general diffusion seing. Recenly, he mehod was furher developed o be applied o he forward backward sochasic differenial equaions FBSDEs. See Fujii and akahashi 212 a,b,c,d, akahashi and Yamada 212, 213 for he deails. Alhough he mehod was exended o be applied o a jump-diffusion model by Kuniomo and akahashi 24 and akahashi27,29, hey concenraed on approximaion of only bond prices or/and plain-vanilla opion prices under a local volailiy jump-diffusion model, and did no derive higher order expansions han he firs order for he opion pricing. Subsequenly, akahashi and akehara 21 found a scheme for pricing plain-vanilla opions in a jump-diffusion wih sochasic volailiy model. However, hanks o a linear srucure of he underlying asse price process in heir model hey separaed he jump componen wih a known characerisic funcion and hen applied he expansion echnique developed in he diffusion models. Hence, heir scheme can no be applied direcly o more general models nor baske opion pricing. he curren work generalizes hese preceding researches in he asympoic expansion approach. he organizaion of he paper is as follows: Afer he nex secion briefly describes our model for baske opions, Secion 3 derives a new approximae pricing formula, and Secions 4 and 5 show numerical examples. Paricularly, Secion 5 provide approximae prices for baske opions on he WI fuures and Bren fuures based on he parameers hrough calibraion o he plain vanilla opion prices. Secion 6 concludes. Appendix shows he derivaion of he coefficiens in he pricing equaion and he condiional expecaion formulas necessary for obaining he main heorem. 2 Model his secion shows he model of he underlying asse prices and heir volailiy processes, which is used for pricing he European ype baske opions. In paricular, suppose ha he filered probabiliy space Ω, F, P, {F } is given, where P is an equivalen maringale measure and he filraion saisfies he usual condiions. hen, S i, ] and σ i, ], i 1,, d represen he underlying asse prices and heir volailiies for, ], respecively. Paricularly, le us assume ha S i and σi are given by he soluions of he following sochasic inegral equaions: S i s i α i S d i ϕ S i σ i, S i dw S i 3

5 N n l, j1 j1 h S i,l,jsτ i j,l Λ l S Eh i S i,l,1]d, 1 ϕ σ i σ i σ i λ i θ i σ d i N n l, h σ i,l,jστ i j,l σ i dw σ i Λ l σ Eh i σ i,l,1]d, 2 where s i and σi, i 1,, d are given as some consans. he noaions are defined as follows: α i i 1,, d are consans. λ i and θ i i 1,, d are nonnegaive consans. ϕ S ix, y and ϕ σ ix are some funcions wih appropriae regulariy condiions. W Si and W σi, i 1,, d are correlaed Brownian moions. Each N l, l 1,, n is a Poisson process wih consan inensiy Λ l. N l, l 1,, n are independen, and also independen of all W Si and W σi. τ j,l sands for he j-h jump ime of N l. Nl, For each l 1,, n and i 1,, d, boh j1 h S i,l,j and Nl, j1 h σ i,l,j are compound Poisson processes. N l, j1 when N l,. For each l and x i, h x i,l,j j N are independen and idenically disribued random variables, where x i sands for one of S i and σ i i 1,, d. for he consan jump case, h x i,l,j H x i,l for some consan H x i,l in all j. for he log-normal jump case, h x i,l,j e Y x i,l,j 1, where Y x i,l,j is a random variable which follows a normal disribuion wih mean m x i,l and variance γ 2 x i,l ha is, Nm x i,l, γ 2 x i,l. h x i,l,j and h x i,l,j h x i,l,j and h x i,l,j l l are independen. j j are independen. N l and h x i,l,j are independen. For he same l and j, h S i,l,j and h σ i,l,j i, i 1,, d are allowed o be dependen, ha is Y S i,l,j and Y σ i,l,j i, i 1,, d are generally correlaed. Remark. By specifying he funcions ϕ S i and ϕ σ i, we can express various ypes of local-sochasic volailiy models. For example, he model wih ϕ S iσ, S as 2 bs c σ and ϕ σ iσ σ corresponds o an exension of he Quadraic Heson model. he model wih ϕ S iσ, S S β Sσ and ϕ σ iσ σ corresponds o an exended SABR λ- SABR model, and he one wih ϕ S iσ, S S β Sσ, ϕ σ iσ σ βσ and λ corresponds o a local volailiy on volailiy wih jumps model. 4

6 3 New Pricing formula for Baske Opion In his secion, we derive an approximaion formula for he baske opion price in he following seps. 1. Inroduce perurbaion parameer ϵ o he model processes, and expand he processes wih respec o ϵ around ϵ as in Proposiion Subsiue he expanded processes for he payoff funcion, and expand he payoff funcion wih respec o ϵ around ϵ. 3. ake he condiional expecaion of each erm in he expanded payoff funcions o calculae analyically he expecaion of he expanded payoff funcions. 4. Use Lemma 3.2 and Appendix B o calculae each condiional expecaion. In he condiional expecaion, each formula in Lemma 3.2 or Appendix B is applied according o he ype of he funcional form of he inegrand, and he calculaion resuls are given in Appendix A. 5. Collecing hese erms in Appendix A wih he same order of he Hermie polynomials, we obain he coefficiens in heorem 3.3. Firs, we inroduce perurbaions o he model 1 and 2. ha is, for a known parameer ϵ, 1] we consider he following sochasic inegral equaions: for i 1,, d, S i,ϵ s i σ i,ϵ σ i N l, n j1 N l, n α i S i,ϵ d ϵ ϕ S i h ϵ S i,l,j Si,ϵ τ j,l σ i,ϵ λ i θ i σ i,ϵ d ϵ ϕ σ i j1 h ϵ σ i,l,j σi,ϵ τ j,l, Si,ϵ Λ l S i,ϵ Ehϵ σ i,ϵ Λ l σ i,ϵ Ehϵ dw Si S i,l,1 ]d dw σi σ i,l,1 ]d, 3. 4 Here, h ϵ x i,l,j ϵh x i,l for all j in he consan jump case. h ϵ x i,l,j eϵy x i,l,j 1, where ϵy x i,l,j Nϵm x i,l, ϵ 2 γ 2 in he log-normal jump case. Noe ha x i h in he,l x i,l,j boh cases. We also define he following perurbed model wih no jump processes, S i,lsv ϵ and σ i,lsv ϵ, which will be used for our approximaion of he baske opion pricing: for i 1,, d, i,lsv ϵ S s i α i S i,lsv ϵ d ϵ 5 ϕ S i σ i,lsv ϵ, S i,lsv ϵ dw Si, 5

7 i,lsv ϵ σ σ i λ i θ i σ i,lsv ϵ d ϵ We assume he asympoic expansions of S i,ϵ ϕ σ i and σ i,ϵ σ i,lsv ϵ dw σi. 6 around ϵ as follows: S i,ϵ S i, ϵs i,1 ϵ2 2! Si,2, 7 σ i,ϵ σ i, ϵσ i,1 ϵ2 2! σi,2, 8 h ϵ h x i,l,j x i,l,j ϵh1 x i,l,j ϵ2 2! h2 x i,l,j, 9 where S i,ι : ι S i,ϵ ϵ ϵ, σ i,ι ι : ισ i,ϵ ϵ ϵ, h ι ι x i,l,j : ι ϵ h x i,l,j ϵ ϵ. ι We also suppose ha W S1,, W Sd, W σ1,, W σd ϱ Z where ϱ is a 2d 2d correlaion marix, and Z is a 2d-dimensional independen Brownian moion. Firsly, we consider a simple case wih one asse and one jump facor, ha is i 1 and l 1 in he above model: S ϵ s N σ ϵ σ j1 N j1 αs ϵ d ϵ ϕ S i h ϵ S,j Sϵ τ j ΛS ϵ σ ϵ, Sϵ dz S Ehϵ S,1 λθ σ ϵ d ϵ h ϵ σ,j σϵ τ j Λσ ϵ We derive S and S 1 explicily. is calculaed as follows: S S s ϵ j1 ϕ S i ϕ σ i Ehϵ σ,1 ]d, 1 α S ϵs1 d σ ϵ dz σ ]d. 11 σ ϵσ1, S ϵs1 dz S N h S,j ϵh1 S,j S τ j ϵs1 τ j s ]d Λ S ϵs1 Eh S,1 ϵh1 S,1 ϵ α S d. 12 6

8 S can be solved as S e α s, and σ θ σ θe λ is derived in he same way. Nex, we calculae S 1. ϵ S 1 Sϵ ϵ ϵ ϵ ϕ S i ϕ S i ϕ S i N j1 j1 α S 1 ϵs2 d σ ϵσ1, S ϵs1 dz S σ 1 ϵσ2, S ϵs1 dz S σ ϵσ1, S1 ϵs2 dz S h 1 S,j ϵh2 S,j S τ j ϵs1 τ j N h S,j ϵh1 S,j S 1 τ j ϵs2 τ j N j1 Λ S 1 ϵs2 Eh S,1 ϵh1 S,1 ]d ]d Λ S ϵs1 Eh 1 S,1 ϵh2 S,1 ϵ α S 1 d ϕ S i h 1 S,j S τ j ΛS σ, S dz S Eh1 S,1 ]d. 13 his equaion can be solved by mehod of variaion of consans as: S i S 1 e α Φ S σ, S dz S N j1 h 1 S,j eα s ΛEh 1 S,1 ]eα s. 14 i 2, 3, can be derived in a similar manner. We explain he muli dimensional case 3, 4, 5, 6 based on hese resuls. For ease of he expressions we inroduce he following noaions: Φ S i,j : ϕ S iσ i, S i ϱ i,j and Φ σ i,j : ϕ σ iσ i ϱ di,j, where ϱ i,j denoes he i, j-elemen of ϱ. 7

9 Φ S i : Φ S i,1,, Φ S i,2d and Φ σ i : Φ σ i,1,, Φ σ i,2d are 2d-dimensional vecors. Φ S : Φ S 1,, Φ S d and Φ σ : Φ σ 1,, Φ σ d are d 2d marices. We define a operaor as follows: When A and B are d 2d marices, A 1,1 B 1,1 A 1,2d B 1,2d A B : A d,1 B d,1 A d,2d B d,2d When A is a d 2d marix and B is a d-dimensional vecor, A 1,1 B 1 A 1,2d B 1 A B B A : A d,1 B d A d,2d B d When A and B are d-dimensional vecors, A B : A 1 B 1. A d B d. 17 We also define x Φˆx x S or σ, ˆx S or σ as x Φˆx : x 1 Φˆx 1,1 x 1 Φˆx 1,2d..... x d Φˆx d,1 x d Φˆx d,2d, 18 where Φˆx i,j denoes he i, j-elemen of he d 2d marix Φˆx. Le us inroduce he following noaions: S S 1,, S d, σ σ 1,, σ d, S j S 1,j,, S d,j, σ j σ 1,j,, σ d,j, h i S,l,j hi S 1,l,j,, hi, S d hi,l,j σ,l,j hi σ 1,l,j,, hi σ d,l,j, e α e α1,, e αd and e λ e λ1,, e λd. Based on hese preparaions, we obain he nex proposiion. 8

10 Proposiion he coefficiens, S i, hi x,l,j x, S, σ, i, 1, 2 and σi i, 1 in he expansions 7, 8 and 9 are given as follows: S e α s, 19 σ θ σ θ e λ, 2 h x,l,j, S 1 σ 1 e α Φ S N n l, j1 e λ Φ σ σ, S dz 21 ] h 1 S,l,j Λ l E h 1 S,l,1 S, 22 σ ] Λ l E h 1 σ,l,1 e λ dz N n l, j1 h 1 σ,l,j e λ τ j,l σ τ j,l e λ σ d, 23 h 1 x,l,j H x,l : H x 1,l,, H x d,l, for all j, consan jump case 24 h 1 x,l,j Y x,l,j : Y x 1,l,j,, Y x d,l,j, log-normal jump case 25 e α S Φ S σ, S S 1 dz S e α σ Φ S σ, S σ 1 dz n N l, N l, j1 j1 ] h 2 S,l,j Λ l E h 2 S,l,1 S ] h 1 S,l,j eα τj,l S 1 τ j,l 2Λ le h 1 S,l,1 e α e α S 1, d h 2 x,l,j R d, for all j, consan jump case 27 h 2 x,l,j Y x,l,j Y x,l,j. log-normal jump case 28 LSV i LSV i 2. he coefficiens, S i 1, 2, 3 and σ expansions of 5 and 6 are given as follows: LSV 1 S LSV 1 σ i 1, 2 in he asympoic e α Φ S σ, S dz, 29 e λ Φ σ σ 9 dz, 3 26

11 LSV 2 S 2 2 LSV 2 σ 2 2 LSV 3 S e α S Φ S σ, S S e α σ Φ S σ, S e λ σ Φ σ e λ σ Φ σ σ σ σ e α SΦ 2 S σ, S e α S Φ S σ, S LSV 1 dz LSV 1 σ dz, 31 LSV 1 dz e λ u Φ σ LSV 1 S S S LSV 2 dz σ u LSV 1 dz u dz, dz e α σφ 2 S σ, S σ LSV 1 σ LSV 1 dz 32 e α σ Φ S σ, S σ LSV 2 dz. 33 Nex, le us define he payoff of a baske call opion wih srike price K as gx K : max{gx K, }, 34 d gx : w x w i x i, where gx represens a weighed sum of he underlying asse prices of x 1,, x d wih he consan boh posiive and negaive weighs w 1,, w d. Here, we se x : x 1,, x d and w : w 1,, w d. For an approximaion of a baske opion price, we firsly noe ha g S ϵ is expanded around ϵ as: g S ϵ g S ϵg S 1 ϵ2 2 g S 2 ϵ3 6 g S 3 oϵ hen, for a srike price K gs ϵy for an arbirary y R, he payoff of he call opion wih mauriy is expanded as follows: g S ϵ K ϵ gs ϵ ϵ g S 1 gs ϵ ϵ 2 g y S 2 ϵ2 6 g ϵ gs 1 y ϵ 2 2 1{ gs 1 1 S 3 > y} g y oϵ 2 S 2

12 1 ϵ 3 6 1{ g gs 1 > y} S δ{ g gs 1 y} oϵ 3 ϵ gs 1 y ϵ 2 2 1{ g S 2 gs 1 > y} 1 ϵ 3 6 1{ LSV 3 g S gs 1 > y} 1 8 δ{ LSV 2 2 g S gs 1 y}. 36 LSV 3 LSV 2 2 2, We apply g S and g S insead of g S 3 and g S 2 which provides reasonable accuracies wih less compuaional burden in he approximaions. We nex noe ha when he number of jumps is k l l 1,, n, ha is on {N l k l } : {N 1, k 1,, N n, k n }, S 1 in he equaion 22 becomes where ξ {kl } Ŝ, ξ {kl } : consan jump Ŝ : S n k l Λ l m S,l e α s, 38 log-normal jump Ŝ : e α Φ S σ, S dz, 39 e α Φ S σ, S dz Here, we use he following noaions: γ S,l γ S 1,l,, γ S d,l and s s 1,, sd n k l γ S,l ζ S,j,l e α s.4 ζ S,j,l ζ S 1,j,l,, ζ S d,j,l and ζ σ,j,l ζ σ 1,j,l,, ζ σ d,j,l are vecors of random variables, where ζ S i,j,l and ζ σ i,j,l follow N, 1, ha is he sandard normal disribuion. ϑ is defined o be he 2d 2d correlaion marix among ζ S i,j,l and ζ σ i,j,l, i 1,, d, hough i does no explicily appear here. We remark ha he disribuion of gŝ is N,, ha is he normal disribuion wih mean zero and variance whose densiy funcion is expressed as { } nx;, 1 x 2 : exp 2π 2Σ {k. 41 l} 11 j1

13 Here, is defined as follows: consan jump : log-normal jump : w e α Φ S σ, S w e α Φ S σ, S d, 42 w e α Φ S σ, S w e α Φ S σ, S d n k l w γ S,l e α s ϑ ζs,l w γ S,l e α s, 43 where ϑ ζs,l sands for he correlaion marix of ζ S,j,l ζ S 1,j,l,, ζ S d,j,l, and x denoes he ranspose of x. Nex, we define η 2 x, {k l } E g S 2 g Ŝ x, {N l k l }], 44 g Ŝ x, {N l k l }] η 3 x, {k l } E η 22 x, {k l } E g g S S LSV 3 LSV 2, 45 2 g Ŝ x, {N l k l }]. 46 Wih hose preparaions, we approximae he expecaion of he baske call payoff under an equivalen maringale measure in he following way: E g ϵe E S ϵ E ϵ2 2 E E ϵ3 6 E E ϵ3 8 E ] K gs 1 ]] g y Ŝ x, {N l k l } 1 { gs 1 1 { gs 1 > y} g > y} g δ { g gs 1 y} S 2 S S g Ŝ x, {N l k l }]] LSV 3 LSV 2 g Ŝ x, {N l k l }]] 2 g Ŝ x, {N l k l }]]. 47 We also noe ha he probabiliy of {N l k l } : {N 1, k 1,, N n, k n } is expressed as p {kl } : n Λ l k le Λ l, 48 k l! which is he produc of he k l imes of he jump probabiliies of N l, l 1,, n, ha is n P {N l, k l }, hanks o he independence of N l, l 1,, n. 12

14 hen, we calculae he coefficiens of ϵ, ϵ2 2, ϵ3 6 and ϵ3 8 on he righ hand of 47 as follows: he coefficien of ϵ is given by: ]] g E E g y Ŝ x, {N l k l } S 1 k n k lk p {kl } gξ {kl }y x gξ{kl } y nx;, dx, 49 where he formula 1 in Lemma 3.2 is used o calculae he condiional expecaion of he second erm on he righ-hand side of he equaion 22 for he consan jump case, and he formula 2 in Lemma 3.2 is used o calculae he condiional expecaion of he second erm of he equaion 22 for he log-normal jump case. he following calculaions for condiional expecaions also use he formulas in Lemma 3.2 or/and Appendix B. he coefficien of ϵ2 2 E E 1 { gs 1 k n k lk he coefficien of ϵ3 6 E E is given by: > y} g p {kl } is given by: 1 { gs 1 k n k lk and he coefficien of ϵ3 8 E E > y} g p {kl } S 2 g Ŝ x, {N l k l }]] gξ {kl }y S is given by: δ { g gs 1 y} k n k lk LSV 3 gξ {kl }y S LSV 2 η 2 x, {k l }nx;, dx, 5 g Ŝ x, {N l k l }]] η 3 x, {k l }nx;, dx, 51 2 g Ŝ x, {N l k l }]] p {kl }η 22 gξ {kl } y, {k l }n gξ {kl } y;,. 52 hen, he iniial value, CK, of he baske call opion wih mauriy and srike K is expanded around ϵ as follows: ] CK, E g K S ϵ k n k lk p {kl }e {ϵ r y {kl } 13 x y {kl } nx;, dx

15 ϵ 2 y {kl } ϵ 3 y {kl } η 2 x, {k l }nx;, dx η 3 x, {k l }nx;, dx ϵ 3 η 22 y {kl }, {k l }n y {kl };, }, 53 where y {kl } : gξ {kl } y, and r is a consan risk-free rae. In order o evaluae η 2 x, {k l }, he condiional expecaions defined in 44, we prepare he following lemma. o evaluae η 3 x, {k l } and η 22 y {kl }, {k l } defined in 45 and 46, respecively, we apply he condiional expecaion formulas for he Wiener-Iô inegrals lised in Appendix B. Lemma 3.2. We suppose he following: W is a d-dimensional Brownian moion. Each N l, l 1,, n is a Poisson process wih inensiy Λ l and hey are independen. τ j,l sands for he ime of he j-h jump in N l. W and N l are independen. X j,l X 1 j,l,, Xd j,l, j 1,, l 1,, n follows a d-dimensional normal disribuion wih mean and variance-covariance marix Θ X,l whose diagonal elemens are 1, ha is each variance is 1. X j,l and X j,l are independen for j j or l l. X j,l are independen of W and N l. Each f 1,l is a d-dimensional vecor in R d. f 2, g 1,l, g 2 and g 2,l l 1,, n are R R d deerminisic funcions and are inegrable wih respec o in he formulas below. For he noaional convenience, f 2, g 1,l, g 2 and g 2,l are expressed as f 2,, g 1,l,, g 2, and g 2,l,, respecively. We define Ŷ and Ŷ as follows: Ŷ : f 2, dw : f Ŷ 2, 2 d n N l, j1 f 1,l X j,l, 54 n k l f1,l Θ X,l f 1,l, 55 where x y sands for he inner produc of x and y in R d, and x denoes he ranspose of x. 14

16 We define I as I 1,, 1. hen, we have he following formulas he proof will be given upon reques N n,l E g 1,l,τj,l I Ŷ y, {N,l k l } j1 N n,l E g 1,l,τj,l X j Ŷ y, {N,l k l } E j1 n k l E E k l g 2, g 2, n E n n k l k l k l n k l g 1,l, Id, 56 H 1 y, g1,l, Θ Ŷ X,lf 1,l d Σ {k, 57 l} Ŷ N n,l g 1,l,τj,l I dw Ŷ y, {N,l k l } j1 g 2, f 2, g 2, I g 2, I n H 1 y, Ŷ g 1,l,s Idsd Σ {k, 58 l} Ŷ N n,l g 1,l,τj,l X j dw Ŷ y, {N,l k l } j1 H 2 y, g 2, f 2, g1,l,s Θ X,lf 1,l dsd Σ {k 2, 59 l} Ŷ Ŷ N n,l g 1,l,τj,l Id Ŷ y, {N,l k l } j1 g 2, I g 1,l,s Idsd, 6 N n,l g 1,l,τj,l X j d Ŷ y, {N,l k l } j1 g 2, I H 1 y, g1,l,s Θ Ŷ X,lf 1,l dsd Σ {k, 61 l} Ŷ 15

17 N n,l E g 1,l,τj,l I j1 n n k l n k l k l N n,l E j1 n n k l n k l τj,l g 2, dw Ŷ y, {N,l k l } H 1 y, Ŷ g 1,l, Id g 2, f 2, d Ŷ H 1 y, Ŷ g 1,l,s Idsg 2, f 2, d Ŷ g 1,l, I k l g 1,l,τj,l H1 y, g 2,s f 2,s dsd Ŷ Ŷ, 62 τj,l g 2, dw X j Ŷ y, {N,l k l } H 2 y, g1,l, Θ X,lf 1,l d g 2, f 2, d Σ {k 2 l} Ŷ H 2 y, g1,l,s Θ X,lf 1,l dsg 2, f 2, d Σ {k 2 l} Ŷ g1,l, Θ X,lf 1,l Ŷ Ŷ H2 y, Ŷ g 2,s f 2,s dsd Σ {k 2, 63 l} Ŷ N n,l E g1,l,τj,l X j g2,l,τj,l X j Ŷ y, {N,l k l } j1 n k l N n,l E g 1,l,τj,l I j2 H 2 y, g 1,l, Θ X,l f1,l g 2,l,Θ X,l f 1,l d Σ {k 2 l} Ŷ n L1 N τj,l,l J1 Ŷ g 2,L,τJ,L I Ŷ y, {N,l k l } g 1,l, g 2,l, d, 64 16

18 n k l k l 1 2 N n,l E g 1,l,τj,l I j2 n k l k l 1 2 N n,l E g 1,l,τj,l I j2 n k l k l 1 2 N n,l E j2 n g 1,l,τj,l k l k l 1 2 where H k x; Ŷ g 1,l, I n L1 g 1,l, I n L1 n L1 N τj,l,l J1 n L1 N τj,l,l J1 g 1,l, Θ X,lf 1,l n L1 N τj,l,l J1 g 1,l, Θ X,lf 1,l g 2,L,s Idsd, 65 g 2,L,τJ,L X J Ŷ y, {N,l k l } H 1 y, g2,l,sθ Ŷ X,L f 1,L dsd Σ {k, 66 l} Ŷ g 2,L,τJ,L X j Ŷ y, {N,l k l } n L1 H 1 y, Ŷ g 2,L,s Idsd Σ {k, 67 l} Ŷ g 2,L,τJ,L X J X j Ŷ y, {N,l k l } n L1 H 2 y, g2,l,sθ X,L f 1,L dsd Σ {k 2, 68 l} Ŷ x, H 2 x; x 2 and H Ŷ Ŷ 4 x; x 4 6 x 2 3 Ŷ Ŷ Ŷ denoes he k-h order Hermie polynomial. Paricularly, H 1 x; 2. Ŷ Applying he above lemma and he condiional expecaion formulas in Shiraya and akahashi 214 which are lised in Appendix B, we obain an approximae pricing formula for a baske call opion wih ϵ 1. he formula for a baske pu opion is easily obained hrough he pu-call pariy. heorem 3.3. An approximaion formula for he iniial value CK, of a baske call opion wih mauriy and srike price K is given by he following equaion: p {kl }e {y r kl C 1,kl N y H kl C 2,kl Σ {k 1 y kl ; l} C 3,kl k n k lk H 2 y kl ; H 4 y kl ; C 4,kl 2 C 5,kl 4 C 6,kl ny kl ;, } Ŷ, 69 17

19 Λ l k l e Λ l where p {kl } n k l!, r is a consan risk-free rae, y gs K, y {kl } gξ {kl } y, Nx denoes he sandard normal disribuion funcion and nx;, Σ 1 2πΣ exp. Here, x 2 2Σ is given by 43, and ξ {kl } is defined by 38. he coefficiens C 1,kl,, C 6,kl are some consans. he derivaion of he coefficiens C 1,kl,, C 6,kl is shown in Appendix A. Moreover, H k x; denoes he k-h order Hermie polynomial: paricularly, H 1 x; x, H 2 x; x 2 and H 4 x; x 4 6 x Numerical Examples his secion shows concree numerical examples based on our mehod developed in he previous secion. 4.1 Seup We apply he following model for numerical experimens under he risk-neural probabiliy measure: each underlying asse price process has a CEV consan elasiciy of variance-ype diffusion erm wih compound Poisson jump componen, and each volailiy process has a CEV-ype diffusion erm wih mean reversion drif and compound Poisson jump componen: S i S i α i Sd i σs i i β S i dw Si N n l, h S i,l,jsτ i j,l Λ l S Eh i S i,l,1]d, 7 j1 σ i σ i λ i θ i σd i N n l, j1 h σ i,l,jστ i j,l ν i σ i β σ i dw σi Λ l σ Eh i σ i,l,1]d, 71 where he jump size h x i,l,j is given by h x i,l,j H x i,l for all j wih a consan H x i,l in he consan jump case, and by h x i,l,j e Y x i,l,j 1 wih Y x i,l,j following a normal disribuion Nm x i,l, γ 2 for all j in he log-normal jump case. x i,l Applying our approximae formula we calculae he baske call opions whose number of he underlying asse is five in he baske. For illusraive purpose we only consider a sysemaic jump case, ha is all he jumps of he underlying asse prices and heir volailiies occur a he same ime i.e. n 1 and ϑ x i,y j 1 where ϑ denoes he 1 1 correlaion marix among ζ S i,j,l and ζ σ i,j,l i 1,, 5, and he inensiy parameer Λ is fixed as 1, hough we are able o rea more general cases. he base 18

20 parameers in he asse price and heir volailiy processes are he same among all he asses, which are lised in he following ables able 1 and able 2. able 1: Common Parameers S i σ i α i β S i β σ i λ i θ i ν i w i Λ n m Si γ Si H σ i able 2: Correlaions S 1 S 2 S 3 S 4 S 5 σ 1 σ 2 σ 3 σ 4 σ 5 S S S S S σ σ σ σ σ Numerical Resuls able 3 shows he resuls for he numerical experimen wih he benchmarks compued by Mone Carlo simulaions, where he number of he ime seps is 128 and he number of rials is 1 million wih aniheic variables in compuaion of each benchmark. We provide a sensiiviy analysis o examine how he approximaion errors by our formula change wih changes in he model parameers. In paricular, we compare he approximaion errors for baske call opion prices wih differen parameers. High or Long means he wice value of he base parameer given in able 1, and Low or Shor means he half value of he base parameer, excep for he correlaion parameer. AE means he asympoic expansion mehod, and MC means he Mone Carlo mehod. 19

21 AE MC Diff Srikes Base Parameers ρ ρ ρ ρ σ High σ Low lambda High lambda Low θ High θ Low ν High ν Low m S High m S Low γ S High γ S Low H σ High H σ Low Λ High Λ Low Mauriy Long Mauriy Shor able 3: AE vs MC I is observed ha he approximaion errors become large when he jump size parameers such as he sandard deviaion of he price s jump size γ S and he volailiy jump size H σ, and he volailiy on he volailiy parameer ν are large. However, in mos of he cases, our approximaion formula works quie well. While in erms of he compuaional ime our analyical mehod is obviously much faser han he Mone Carlo simulaions wih 128 ime seps and one million rials. 5 WI - Bren baske opions his secion presens numerical examples for pricing WI - Bren baske opions based on our approximaion scheme wih he parameers obained by calibraion o he acual fuures opion prices. In paricular, we use he following model under he risk-neural probabiliy measure, where each underlying asse price process has a CEV consan elasiciy of variance-ype diffusion erm wih compound Poisson jump componen and each volailiy process follows a log-normal model: for i 1, 2, S i S i σs i i β i dw Si σ i σ i ν i σdw i σi, N n l, j1 h S i,l,jsτ i j,l Λ l S Eh i S i,l,1]d, where he jump size in he fuures price process is log-normally disribued, ha is h S i,l,j e Y S i,l,j 1 wih Y S i,l,j following a normal disribuion Nm S i,l, γ 2 for all j. S i,l Applying our approximae formula o his model, we calculae he baske opions on WI fuures and Bren fuures. For simpliciy, we only consider a sysemaic jump case, ha is all he jumps of he underlying asse prices i.e. n 1 and ϑ S i,s j 1 where ϑ denoes he 2 2 correlaion marix among ζ S i,j,l i 1, 2. 2

22 We se he calculaion dae for baske opion prices on March 31, 215. In able 4, we repor he arge baske prices wih heir underlying fuures prices on he dae, he erms o mauriies and he relevan risk-free ineres raes. Asse Price Mauriy Risk Free Rae JUN15 WI % Bren % baske % DEC15 WI % Bren % baske % able 4: Asse Price, Mauriy and Risk Free Rae We firsly need o obain he model parameers hrough calibraion o he relevan opion prices of WI fuures and hose of Bren fuures. In he jump componen, he inensiy parameer Λ is fixed as 1. he oher jump parameers are assumed o ake common values for he wo relevan fuures price processes used for he calculaion of a baske opion price. For compuaional efficiency, he selemen prices of American opions are ransformed o hose of he European opions before calibraion: More precisely, afer an implied volailiy of each American opion price is esimaed under a binomial version of he Black-Scholes model, he corresponding European opion price is compued. Hereafer, his European opion price is called he ransformed CME or ransformed ICE opion price. hen, calibraion is implemened agains he ransformed CME or ransformed ICE opion prices wih differen srikes simulaneously, where ou-ofhe-money OM prices are used for he calibraion; for JUN15 fuures opions, he srikes of he opions range usd 35 o usd 75 wih every five dollars, and for DEC15 fuures opions, hose of he opions range usd 4 o usd 8 wih every five dollars. Moreover, he correlaions beween he fuures prices and heir volailiies are assumed o ake common values for he wo relevan fuures, which are used o calculae a baske opion. hese correlaions are obained by calibraion o he marke fuures opion prices, which are shown in he ρ-column of ables 5 and 6. he correlaions beween he wo fuures price processes are esimaed by he pas hree-monh s hisorical daa of he fuures prices. he correlaions beween he corresponding volailiy processes of he wo fuures prices are assumed o be he same as he correlaions of he fuures prices. hen, we obain he following esimaes: he correlaion beween he WI and Bren fuures prices for JUN15 is.955, and he correlaion for DEC15 is.975. Given he above assumpions in he calibraion, we compare he following wo specificaions of he model: i SABR ype Local sochasic volailiy model wihou jump LSV model 21

23 ii SABR ype Local sochasic volailiy wih log-normal jumps in he fuures prices model LSV jump diffusion model he parameers obained by calibraion o he JUN 15 opion prices of WI and Bren fuures are shown in able 5. σ β ν ρ m S γ S LSV WI % Bren % LSV jump WI % Bren % able 5: Parameers on JUN15 he parameers obained by calibraion o he DEC 15 opion prices of WI and Bren fuures are shown in able 6. σ β ν ρ m S γ S LSV WI % Bren % LSV jump WI % Bren % able 6: Parameers on DEC15 he resuls for he calibraion o he JUN 15 opion prices of WI and Bren fuures are shown in ables 7 and 8. Srike ransformed CME LSV LSV jump Diff LSV Diff LSV jump able 7: JUN15 WI Srike ransformed ICE LSV LSV jump Diff LSV Diff LSV jump able 8: JUN15 Bren he resuls for he calibraion o he DEC 15 opion prices of WI and Bren fuures are shown in ables 9 and 1. 22

24 Srike ransformed CME LSV LSV jump Diff LSV Diff LSV jump able 9: DEC15 WI Srike ransformed ICE LSV LSV jump Diff LSV Diff LSV jump able 1: DEC15 Bren We can observe ha he LSV jump diffusion model gives much beer fiing o he fuures opions han he LSV model. Figures 1 and 2 show he implied volailiies of DEC 15 WI and Bren fuures. Figure 1: Implied volailiies of DEC 15 WI fuures 45% 43% 41% 39% 37% 35% Original LSV LSV jump 33% 31%

25 Figure 2: Implied volailiies of DEC 15 Bren fuures 46% 44% 42% 4% 38% 36% Original LSV LSV jump 34% 32% Especially in OM, LSV model is no able o duplicae he implied volailiies. Using he parameers obained hrough he calibraion, ables 11 and 12 show he comparison of he baske opion prices given by our approximaion AE, and Mone Carlo simulaions MC in he LSV Jump diffusion model. Srike AE MC Diff able 11: JUN15 baske opion price Srike AE MC Diff able 12: DEC15 baske opion price hese resuls show our approximaion formula works well. 24

26 6 Conclusion We have derived a new approximaion formula for baske opion pricing in a model wih local-sochasic volailiy and jumps. In paricular, our model admis a local volailiy funcion and jumps in boh he underlying asse price and is volailiy processes. hanks o he closed form formula he compuaional speed of he mehod is much faser han he oher numerical schemes. Moreover, in numerical experimens, we firsly calibrae he model o he opions on he WI and Bren fuures by applying our approximaion formula for he plainvanilla opion. hen, by using he calibraed parameers, we approximae he prices of he baske opions on he WI and he Bren fuures and compare hose wih he benchmark prices obained by he Mone Carlo mehod, which has demonsraed he effeciveness of our approximaion scheme. We also noe ha he higher order expansions can be derived in he similar manner, which is expeced o provide more precise approximaions as in he diffusion cases in Shiraya, akahashi and oda 212 and akahashi, akehara and oda 212. References 1] Andreasen, J. and Huge, B., ZABR - Expansions for he Mass -, Preprin, ] Bayer,C., and Laurence, P., Asympoics beas Mone Carlo: he case of correlaed local vol baskes, Comm. Pure Appl. Mah., 214, 67: DOI: 1.12/CPA ] Brigo, D., Mercurio, F., Rapisarda, F. and Scoi, R., Approximaed Momen- Maching for Baske-opions Pricing. Quaniaive Finance, 4, 24, ] G. Deelsra, J. Liinev and M. Vanmaele, Pricing of arihmeic baske opions by condiioning, Insurance Mahemaics and Economics, 24, vol. 34, issue 1, pages ] Eraker, B., Do Sock Prices and Volailiy Jump? Reconciling Evidence from Spo and Opion Prices, he Journal of Finance, Vol.LIX, No.3, 24, ] Fujii, M., akahashi, A., Analyical Approximaion for Non-linear FBSDEs wih Perurbaion Scheme, Inernaional Journal of heoreical and Applied Finance Vol.15-5, ] Fujii, M., akahashi, A., Perurbaive Expansion of FBSDE in an Incomplee Marke wih Sochasic Volailiy, Quarerly Journal of Finance Vol.2, No.3, ] Fujii, M., akahashi, A., Perurbaive Expansion echnique for Non-linear FBSDEs wih Ineracing Paricle Mehod, CARF-F-278, ] Fujii, M., akahashi, A., An FBSDE Approach o American Opion Pricing wih an Ineracing Paricle Mehod, CARF-F-32,

27 1] Hagan, P.S., Kumar, D., Lesniewskie, A.S., and Woodward, D.E., Managing Smile Risk, Willmo Magazine, 22, ] Kuniomo, N. and akahashi, A., Pricing Average Opions, Japan Financial Review, Vol.14, 1992, 1-2in Japanese. 12] Kuniomo, N. and akahashi, A., he Asympoic Expansion Approach o he Valuaion of Ineres Rae Coningen Claims, Mahemaical Finance, 11, , ] Kuniomo, N. and akahashi, A., Applicaions of he Asympoic Expansion Approach based on Malliavin-Waanabe Calculus in Financial Problems, Subsequenly published in S. Waanabe eds., Sochasic Processes and Applicaions o Mahemaical Finance, World Scienific, , ] Pagliarani S., Pascucci A., Local sochasic volailiy wih jumps, Inernaional Journal of heoreical and Applied Finance, 17 1, ] Rogers, L.C.G., and Shi, Z., he value of an Asian opion, Journal of Applied Probabiliy, 32, , ] Shiraya, K. and akahashi, A., Pricing Muli-Asse Cross Currency Opions, Journal of Fuures Markes, 34, 1, 214, ] Shiraya, K. and akahashi, A. and oda,m., Pricing Barrier and Average Opions under Sochasic Volailiy Environmen, Journal of Compuaional Finance vol.15-2, 211/12, ] Shiraya, K,, akahashi, A., Yamazaki, A., Pricing Swapions under he LIBOR Marke Model of Ineres Raes wih Local-Sochasic Volailiy Models, Willmo Magazine,volume 212, issue 61, pp.48-63, Sepember, ] akahashi, A., Essays on he Valuaion Problems of Coningen Claims, Unpublished Ph.D. Disseraion, Haas School of Business, Universiy of California, Berkeley, ] akahashi, A., An Asympoic Expansion Approach o Pricing Coningen Claims, Asia-Pacific Financial Markes, Vol. 6, 1999, ] akahashi, A., On an Asympoic Expansion Approach o Numerical Problems in Finance, Seleced Papers on Probabiliy and Saisics, pp , 29, American Mahemaical Sociey. 22] akahashi, A., An Asympoic Expansion Approach in Finance, CARF-F- 12CIRJE-F-59, 27.8revised in ] akahashi, A., akehara, K., A Hybrid Asympoic Expansion Scheme: an Applicaion o Long-erm Currency Opions, Inernaional Journal of heoreical and Applied Finance, Vol.13-8, 21,

28 24] akahashi, A., akehara, K., oda, M., Compuaion in an Asympoic Expansion Mehod, CARF-F-149 CIRJE-F-621, ] akahashi, A., akehara, K., oda, M., A General Compuaion Scheme for a High-Order Asympoic Expansion Mehod, Inernaional Journal of heoreical and Applied Finance Vol.15-6, ] akahashi, A., Yamada,., An Asympoic Expansion for Forward-Backward SDEs: A Malliavin Calculus Approach, CARF-F-296, ] akahashi, A., Yamada,., On an Asympoic Expansion of Forward-Backward SDEs wih a Perurbed Driver, CARF-F-326, ] Xu, G., Zheng, H., Baske Opions Valuaion for a Local Volailiy Jump-Diffusion Model wih he Asympoic Expansion Mehod, Insurance Mahemaics and Economics, 473, 21, ] Xu, G., Zheng, H., Lower Bound Approximaion o Baske Opion Values for Local Volailiy Jump-Diffusion Models, Inernaional Journal of heoreical Applied Finance, 16 8, ] Yoshida, N., Asympoic Expansions for Saisics Relaed o Small Diffusions, Journal of Japan Saisical Sociey, Vol.22, 1992, A Derivaion of Coefficiens his secion derives he coefficiens, C i,kl, i 1,, 6 in he expansion formula 69 in heorem 3.3 under a log-normal jump case. A consan jump case is obained in a similar way. In he following we omi some noaions for simpliciy. Firsly, le us show he expressions of gs 1 gs 1 g 1 g 2! S2 g g n e α Φ S dz N l, j1 n e α τ j,l S τ j,l h 1 S,l,j Λ l Eh 1 S,l,1 ] eα g e α S Φ S g e α S Φ S g e α S Φ S and g 1 2! S2 e α S d e α u Φ S dz u dz j1 : 72 73, 74 N n l, h 1 S,l,j eα τ j,l S τ j,l dz n Λ l Eh 1 S,l,1 ] 27 e α u S u dudz

29 g e α σ Φ S g e α σ Φ S g e α σ Φ S g g g g g g g g n N l, j1 n n j1 n e λ u Φ σ dz u dz N n l, h 1 σ,l,j e λ τ j,l σ τ j,l dz j1 n h 2 S,l,j eα τ j,l S τ j,l Λ l Eh 2 S,l,1 ] eα N l, N l, j1 n j1 N l, n n n Λ l Eh 1 σ,l,1 ] e λ e α S d τj,l h 1 S,l,j eα τj,l e ατj,l u Φ S dz u h 1 S,l,j eα τ j,l h 1 S,l,j eα τ j,l n L1 n L1 Λ l Eh 1 S,l,1 ] eα Λ l Eh 1 S,l,1 ] eα Λ l Eh 1 S,l,1 ] eα N L,τj,l m1 e λu σ u dudz h 1 S,L,m eατ j,l τ m,l S τ m,l Λ L Eh 1 S,L,1 ] eατ j,l n e α u Φ S dz u d N L, L1 m1 n L1 τj,l e αu S u du h 1 S,L,m eα τ m,l S τ m,l d Λ L Eh 1 S,L,1 ] eα e αu S u dud , 88 where E h ϵ E h E x i,l,1 x i,l,1 h 1 x i,l,1 E h 2 x i,l,1 ] ] ] ] E E h ϵ ϵ E ] e ϵy x i,l,1 1 e ϵm x i,l 1 2 ϵ2 γ 2 x i,l 1, 89 x i,l,1] ϵ h ϵ x i,l,1 ϵ 2 E h ϵ x i,l,1 1 1, 9 ] ϵ m x i,l ϵγ 2 x i,l eϵm x i,l 1 2 ϵ2 γ 2 x,l i m x i ϵ,l, 91 ] ϵ γ 2 x i,l eϵm x i,l 1 2 ϵ2 γ 2 x i,l m x i,l ϵγ 2 x i,l 2 e ϵm x i,l 1 2 ϵ2 γ 2 x,l i m 2 x i,l γ2 x i,l. 92 ϵ 28

30 Nex, we define he expression F X as ] F X : E X gŝ x, {N l k l } ] g E X Ŝ x, {N l k l }, 93 where X sands for he expression in he equaion number X. We also define as : d w i e αi Φ S i n k l d w I e αi Φ S d I I1 d w i γ S i,le αi s i d w I ϑ S i,s I γ S I,le αi s I. 94 I1 hen, we obain he following calculaions by using Lemma 3.2 and Appendix B. F gs 1 F 72 F 73 F , 95 F gs 2 F 75 F 76 F 77 F 78 F 79 F 8 F 81 F 82 F 83 F 84 F 85 F 86 F 87 F 88 F 75 F 16 F 17 F 77 F 78 F 79 F 8 F 114 F 115 F 116 F 82 F 121 F 122 F 125 F 126 F 127 F 128 F 85 F 86 F 135 F 136 F , 96 where sand for he equaion numbers lised below. Moreover, F 75 15, F 77 11, F , F , F 8 113, F 82 12, F , F , F , and F 76 F 16 F , 97 F 81 F 114 F 115 F , 98 F 83 F 121 F , 99 F 84 F 125 F 126 F 127 F , 1 F 87 F 135 F H k x; sands for he k-h order Hermie polynomial. ] g F 72 E g e α Φ SdZ Ŝ x, {N l k l } 29

31 d w i e αi Φ S i F 73 E g d d d F 74 E d n w i w i w i g N l, j1 n n n n w i n d w Ie αi Φ S I d H 1x,, 12 I1 h 1 S,l,j eα τ j,l S ] e αi s i E k l Y S i,l,1 gŝ x g τ j,l Ŝ x, {N l k l } e αi s i ] E k l ms i,l γ S i,lζ S,1,l g Ŝ x e αi s i k l m S i,l γ S i,l F 75 E g e α S Φ S d I1 w Iϑ S i,s I γ S I,le αi s I H 1 x, g Λ l Eh 1 S,l,1 ] eα e α S d Ŝ x, {N l k l } ], 13 Λ l m S i,le αi s i, 14 d w i e αi S iφ S i e α u Φ S dz u dz g Ŝ x, {N l k l } d w Ie αi Φ S I e αi u Φ S i N n l, F 76 E g e α S Φ S h 1 S,l,j eα τ j,l S F 16 I1 j1 d J1 ] w Je αj u Φ S J dud H 2x,, 15 2 g τ j,l dz Ŝ x, {N l k l } N n l, g E g e α SΦ S m S,l γ S,l ζ S,j,l e α τj,l S τ j,l dz Ŝ x, {N l k l } j1 N n l, g E g e α S Φ S m S,l e α τj,l S τ j,l dz Ŝ x, {N l k l } 16 j1 N n l, g E g e α S Φ S γ S,l ζ S,j,l e α τj,l S τ j,l dz Ŝ x, {N l k l }, 17 d w i e αi S iφ S i n d F 17 E w i e αi S iφ S i n N l, Nl, k l j1 j1 k l m S i,ls i d w I e αi Φ S I d H1x, Σ{k l}, 18 I1 d w Ie αi Φ S I I1 γ S i,le αi τ j,l S i, τ j,l J1 d d n w i e αi S iφ S i w I e αi Φ k l S I γ S i,ls i I1 n F 77 E g e α S Φ S Λ l Eh 1 S,l,1 ] d w J ϑ S i,s J γ S J,le αj s J d H 2x, {N l k l } d J1 2 w J ϑ S i,s J γ S J,le αj s J d H 2x, e α u S u dudz g Ŝ x, {N l k l } 2 ], 19 3

32 d w i e αi S iφ S i d w Ie αi Φ S I I1 F 78 E g e α σφ S d w i e αi σ iφ S i n Λ l m S i,ls i d H 1x, e λ u Φ σdz udz g Ŝ x, {N l k l } d w Ie αi Φ S I e λi u Φ σ i N n l, F 79 E g e α σφ S h 1 σ,l,j e λ τ j,l σ d w i e αi σ iφ S i I1 j1 d w Ie αi Φ S I I1 d w i e αi σ iφ S i n d w I e αi Φ S I I1 n k l γ σ i,le λ i u σu i, n F 8 E g e α σφ S d w i e αi σ iφ S i F 81 E g F 114 F 115 F 116 E g E g E g d d d F 82 E d n N l, j1 n w i w i w i g N l, j1 n N l, j1 n j1 n n n n w i n N l, d J1 d J1, 11 ] w Je αj u Φ S J dud H2x, Σ{k l}, g τ j,l dz Ŝ x, {N l k l } k l m σ i,le λ i u σ i, u dud H 1x, w Jϑ S i,σ J γ S J,le αj s J dud H 2x,, ] Λ l Eh 1 σ,l,1 ] e λ d w I e αi Φ S I I1 h 2 S,l,j eα τ j,l S n Λ l m σ i,l e λu σ u dudz {Nl k l } g τ j,l Ŝ x, {N l k l } e λi u σ i, u dud H 1x,, 113 g m S,l m S,l e α τj,l S τ j,l Ŝ x, {N l k l } 114 g 2m S,l γ S,l ζ S,j,l e α τj,l S τ j,l Ŝ x, {N l k l } 115 g γ S,l γ S,l ζ S,j,l ζ S,j,l e α τj,l S τ j,l Ŝ x, {N l k l }, 116 k l m 2 S i,l eαi s i, k l 2m S i,lγ S i,le αi s i d I1 k l γ S i,l 2 e 2αi s i 2 w Iϑ S i,s I γ S I,le αi s I H 1x, d n d w i k l γ S i,le αi s i I1 117, 118 w I ϑ S i,s I γ S I,le αi s I ] g Λ l Eh 2 S,l,1 ] eα e α S d Ŝ x, {N l k l } 2 H2 x, 2, 119 Λ l m 2 S i,l γ2 S i,l eαi s i, 12 31

33 F 83 E g F 121 F 122 E g n N l, j1 n N l, j1 d n w i m S i,l d n w i m S i,l d F 84 E g F 125 F 126 n w i γ S i,l I1 d n d w i γ S i,l I1 E g E g F 127 E g E g E g e α s e α s e α s e α s e α s e α s τj,l m S,l e α τj,l τj,l γ S,l ζ S,j,l e α τj,l g e ατj,l u Φ S dz u Ŝ x, {N l k l } 121 g e ατj,l u Φ SdZ u Ŝ x, {N l k l }, 122 d e αi Φ S i w Ie αi Φ S I d k l H 1x, I1 d e αi Φ S i w I e αi Φ S id k l H 1 x,, 123 I1 d d w I ϑ S i,s I γ S I,le αi s I e αi Φ S iw J e αj Φ S J d k l H 2x, J1 2 k n l j l 1 w Iϑ S i,s I γ S I,le αi s I h 1 S,l,j l n j L 1 L1 I1 h 1 k n l m S,l γ S,l ζ S,jl,l j l 1 k d n l w i e αi s i k d n l w ie αi s i k d n l w i e αi s i k d n l w i e αi s i k d n l w ie αi s i k n l n j L 1 j l 1 L1 I1 k n l n j L 1 j l 1 L1 I1 k n l n j L 1 j l 1 L1 I1 k n l n j L 1 j l 1 L1 I1 n j L 1 j l 1 L1 I1 n j L 1 j l 1 L1 I1 n j L 1 j l 1 L1 I1 n j l 1 L1 I1 n j L 1 j L 1 j l 1 L1 I1 d e αi Φ S iw Je αj Φ S J d k l J1 g S,L,I Ŝ x n j L 1 L1 I1 g m S,L γ S,L ζ S,I,L Ŝ x H 2 x, 2, 124 g m S,l m S,L Ŝ x 125 g m S,l γ S,L ζ S,I,L Ŝ x 126 g γ S,l ζ S,jl,l m S,L Ŝ x 127 g γ S,l ζ S,jl,l γ S,L ζ S,I,L Ŝ x, 128 m S i,lm S i,l, 129 ] g m S i,le γ S i,lζ S,I,L Ŝ x m S i,lγ S i,l J1 m S i,le d w J ϑ S i,s J γ S J,Le αj s J H 1x, γ S i,lζ S,jl,l m S i,lγ S i,l J1 ] gŝ x d w Jϑ S i,s J γ S J,le αj s J H 1 x,, 13,

34 F 128 k d n l w i e αi s i k d n l w ie αi s i F 85 E g γ S i,l J1 n n j L 1 j l 1 L1 I1 n j L 1 j l 1 L1 I1 d w J ϑ S i,s J γ S J,Le αj s J N l, j1 h 1 S,l,j eα τ j,l ] g E γ S i,lζ S,jl,lγ S i,lζ S,I,L Ŝ x γ S i,l ι1 d w ιϑ S i,s ιγ S ι,le αι s ι H 2 x, n L1 n N l, E g m S,l γ S,l ζ S,j,l e α d F 86 E E w i d g g j1 n m S i,le αi w i n n n γ S i,le αi n d w i F 87 E g F 135 n L1 Λ Lm S i,ls i k l 2 2, 132 Λ L Eh 1 S,L,1 ] τj,l L1 d w I ϑ S i,s I γ S I,le αi s I I1 Λ l Eh 1 S,l,1 ] eα g e ατj,l u S u du Ŝ x, {N l k l } n τj,l g Λ L m S,l e αu S u du Ŝ x, {N l k l } n Λ L m S i,ls i k l H 1 x,, ] L1 Λ l Eh 1 S,l,1 ] eα d e α Φ S dz e α u Φ S dz u d g Ŝ x, {N l k l } Λ l Eh 1 S,l,1 ] eα du e α Φ S dz g Ŝ x, {N l k l } n Λ l m S i,l,je αi e αi Φ S i d n w i Λ l m S i,l,je αi Φ S i E g d w i d w i F 136 E g E g n s n Λ l m S,l e α n n Λ l m S i,l e αi n Λ l m S i,le αi s i Λ l m S i,le αi s i n L1 n Λ l m S,l e α n Λ l m S,l e α N L, L1 I1 n d w I e αi Φ S I d H1x, Σ{k l} d w Ie αi Φ S I d H 1x,, 134 I1 I1 L1 I1 n m S i,l L1 g m S,L s d Ŝ x, {N l k l } 135 N L, ] g γ S,L ζ S,j,L s d Ŝ x, {N l k l }, 136 k l dud k l m S i,l 2, 137 n L1 N L, NL, k L I1 ] g g E γ S,L ζ S,j,L Ŝ x s d Ŝ x, {N l k l } 33

35 F 88 n L1 d w i s i N L, NL, k L I1 d n w is i Λ l m S i,le αi d w i d w i n ] d γ S,L w J ϑ S i,s J γ S J,Le αj s J H 1 x, g d Ŝ J1 Σ {k x, {N l k l } l} k n n L d Λ l m S i,le αi γ S i,l w J ϑ S i,s J γ S J,Le αj s J H 1x, d L1 I1 J1 n Λ l m S i,le αi Λ l m S i,le αi n γ S i,l L1 J1 n L1 d w Jϑ S i,s J γ S J,Le αj s J n Λ L m S i,ls i d L1 k l 2 H 1 x,, 138 Λ Lm S i,ls i Nex, le us show he expression of g ! SLSV by applying Appendix B g SLSV g e α S Φ S e α u Φ S dz u dz 2! 2 e α σ Φ S e λ u Φ σ dz u dz 2 g e α S Φ S e α u Φ S dz u dz 14 2g e α S Φ S e α σ Φ S g e α σ Φ S e α u Φ S dz u dz e λ u Φ σ dz u dz 141 e λ u Φ σ dz u dz F 14 2F 141 F hen, we obain he expressions of F M for M 14, 141, 142 as follows: d d F M w I q M,3,iq M,1,i q M,2s,iq M,1s,i dsd w i I1 r q M,5r,Iq M,1r,I q M,4u,Iq M,1u,I dudr d w i H 4 x; 4 { d r w I q M,3,iq M,1,i q M,5r,Iq M,1r,I q M,2u,iq M,4u,I dudrd I1 r q M,5,Iq M,1,I q M,3r,iq M,1r,i q M,2u,iq M,4u,I dudrd r q M,3,iq M,1,i q M,2r,iq M,5r,I q M,4u,Iq M,1u,I dudrd 34

36 where q 14,1,i d q M,3,iq M,5,I q M,2s,iq 1s ds q M,4u,Iq M,1u,I du d r } u q M,5r,Iq M,1r,I q M,3u,iq M,4u,I q M,2s,iq M,1s,i dsdudr w i d w I I1 H 2 x; 2 q M,2u,iq M,4u,I duq M,3,iq M,5,I d, 144 d w i Φ S i, 145 q 14,2,i e αi Φ S i, 146 q 14,3,i e αi S iφ S i, 147 q 14,4,i e αi Φ S i, 148 q 14,5,i e αi S iφ S i, 149 d q 141,1,i w i Φ S i, 15 q 141,2,i e αi Φ S i, 151 q 141,3,i e αi S iφ S i, 152 q 141,4,i e λiu Φ σ i, 153 q 141,5,i e αi λ i σ iφ S i, 154 d q 142,1,i w i Φ S i, 155 q 142,2,i e λiu Φ σ i, 156 q 142,3,i e αi λ i σ iφ S i, 157 q 142,4,i e λiu Φ σ, 158 q 142,5,i e αi λ i σ iφ S i. 159 hen, we show he expressions of g by applying Appendix B. g 1 3 SLSV 3! 1 3 3! SLSV 1 2 g e α SΦ 2 S e α u Φ SdZ u dz 2 g e α S Φ S e α u S Φ S u e αu v Φ S dz v dz u dz