APPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14

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1 APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS DUPON OLIVIER Prema 14 Contents Introducton 1 1. Framewor Baset optons 1.. Asan optons. Computng the prce 3. Lower bound 3.1. Closed formula for the prce 3.. Implementaton Computng the deltas 4 4. Upper bound Addtonal defntons and computaton Closed formula Implementaton Computng the deltas 6 References 7 Introducton Routnes lowlnearprce and uplnearprce compute lower and upper bound approxmatons proposed by [1 for the prce of a Call or a Put opton wrtten on a lnear combnaton of Blac-Scholes asset prces. hese routnes also gve approxmatons of the deltas of the clam. Routnes lower_baset upper_baset lower_asan and upper_asan barely use these two general routnes and just correctly ntalze ther parameters. Note that the lower bound approxmaton s far better than the upper bound one so the former s the one to be prefered. 1. Framewor More precsely routnes lowlnearprce and uplnearprce compute lower and upper bound approxmatons of p = E + where : X = ε x e G VarG/ 1 wth G 0 n a centered Gaussan vector of covarance matrx Σ ε = ±1 and x > 0. One may assume that j G G j ; otherwse one can group the terms wth the same Gaussan random varable n the summaton above. If the ε s are all equal to +1 then EX + = EX and f they are all equal to 1 then EX + = 0. We therefore assume from now on that all the ε s are not equal.

2 DUPON OLIVIER For a good choce of the parameters ε x and G p s the prce of a baset or a dscrete tme average Asan opton Baset optons. In the case of a baset opton characterzed by the wegths w appled to the dfferent assets whose volatltes are stored n vector σ and prces at are gven by S 0e G VarG/ r q the notaton means : ε = sgnw x = w S 0e q q beng the dvdend on stoc. he stre K s ncluded n the notaton as stoc 0 : ε 0 = 1 x 0 = K e r r beng the rate of nterest and σ 0 = 0. he covarance matrx of vector G 0 n then s : Σ j = σ σ j C j where C j s the correlaton between stocs and j. he prce p then s gven by : n + p = e r E w S K. =1 1.. Asan optons. As for dscrete-tme average Asan optons over n equally spaced dates the notaton wll amount to the followng : ε 0 = 1 x 0 = K e r and ε = 1 x = 1 n S0 er q n r q s the dvdend yeld on the stoc. σ beng the volatlty of the stoc the covarance matrx of vector G s gven by Σ j = mnj n σ.. Computng the prce Wth the prevous notaton prce p s gven by p = EX +. Lower and upper bounds derve from the followng observaton : sup EXY = EX + = 0 Y 1 nf X=Z 1 Z Z 1 0 Z 0 EZ 1 where X Y Z 1 and Z are random varables. Indeed for 0 Y 1 EXY = EX + Y EX Y EX +. And for Y = 1 {X 0} the supremum s attaned. Moreover f X = Z 1 Z wth Z 1 and Z postve Z 1 X + leadng to EZ 1 EX +. And for Z 1 = X + X = X + X the nfmum s attaned. 3. Lower bound 3.1. Closed formula for the prce. A closed formula s obtaned for the lower bound by restrctng the supremum n over {Y u R n+1 and d R Y = 1 {u G d} }. Lettng p = supex1 u G d ud and rewrtng σ = VarG = Σ one gets the computatonally effcent formula : p = sup sup ε x Φd + σ Cv where d R =1 Φx = 1 x u e π denotes the cumulatve dstrbuton functon of the Normal law; C s the correlaton matrx C j = Σj σ σ j ; C s such as C C = C.

3 APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS 3 Proof : By condtonng and lnearty p = sup sup ε x E Ee G VarG / u G1 {u G d}. d R u R n+1 Snce G u G forms a centered Gaussan vector EG u G = CovG u G u G and Varu G Ee G VarG / u G = e EG u G VarEG u G/ Snce Varu G = u Σu CovG u G p = sup ε x E exp u G CovG u G / 1 {u G d} ud u Σu u Σu = sup sup ε x Ee CovG u Gu G CovG u G / 1 {u G d} d R u Σu=1 = sup sup d R u Σu=1 = sup sup d R u Σu=1 ε x Ee Σu u G Σu / 1 {u G d} ε x Φd + Σu hen defnng D as the dagonal matrx wth dagonal coeffcents σ Σ = D C C D so that u Σu = u D C C Du = C Du = 1 tang v = C Du leads to Σu = D Cv = σ Cv and fnally : p = sup sup ε x Φd + σ Cv d R =1 3.. Implementaton. he goal of routne lowlnearprce s therefore to compute the maxmum of the functon v d n ε x Φd + σ Cv under the constrant = 1. Rather than optmzng under ths constrant routne lowlnearprce computes the unconstraned maxmum of the functon Fv d = n ε Cv x Φd + σ. he lower bound approxmaton of the prce wll therefore be : p = ε x Φ d Cv + σ v 3 where d and v are the soluton of the unconstraned problem. Optmzaton n lowlnearprce routne uses a smple conjugate gradent method. Frst-order dervatves must therefore be nown. One can chec that : wth F v j = ε x σ ϕ d + σ Cv Cj v j Cv F n d = ε x ϕ d + σ Cv ϕx = 1 e x π he matrx C whch s a parameter of ths problem s computed by Cholesy decomposton and Φ s obtaned thans to the ncomplete Gamma functon tself computed as a seres. he other parameters ε x σ and dmenson n are nown cf. 1 so there s no need for further computaton to mplement the algorthm.

4 4 DUPON OLIVIER Fnally note that the correlaton between two dstnct stocs s not necessarly constant n routnes lowlnearprce and uplnearprce but t s supposed so n routnes that specfcally prce baset or Asan optons Computng the deltas. he pont d v where the functon F reaches ts maxmum depends on the x s. But because of the Euler equatons of optmalty for d and v one smply has : x = ε Φ d + σ Cv Baset optons. In the case of baset optons x = w S 0e q for > 0 thus leadng to : δ = S 0 = x x S 0 = ε Φ d Cv + σ x S Asan optons. In ths case x = 1 n S0 er q n r and n S0 = p x [ = ε Φ d Cv + σ x S0 =1 =1 4. Upper bound x S Addtonal defntons and computaton. For 0 n let σ = Σ Σ + Σ = VarG G. σ = 0 only for =. hen choosng λ such as λ = ε and λ ε > 0 for all t s possble because all the ε do not have the same sgn X n 1 can be rewrtten as : X = ε x e G VarG/ λ x e G VarG / 4 for every = 0... n. 4.. Closed formula. A closed formula s obtaned for the upper bound by restrctng the nfmum n over { ε x e G VarG/ λ x e G VarG / } wth the same notatons as n 4. Lettng p = mn 0 n nf λ = ε E ε x e G VarG/ λ x e G the effcent computatonal formula s : n p = mn ε x Φd + ε σ 0 n wth d beng the one soluton of ε x ϕd + ε σ = 0 Proof : Prce s gven by : p = mn 0 n nf λ = ε E [ VarG / ε x e G VarG / λ x e G VarG /

5 APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS 5 and [ E ε x e G VarG / λ x e G + VarG / = E [ e G VarG / ε x e G G VarG VarG / λ x = E [ ε x e G G σ σ / λ x Under probablty P gven by dp = dp eg VarG / G σ σ = g N 0 1. Lettng g = G σ dstrbuted under P and : Now Hence Σ σ σ g Σ σ g and g are ndependent and are both normally G G σ σ = wth g N 0 1 and hs leads to : σ g σ Σ σ σ g. Var σ g σ Σ g = σ + σ Σ σ = σ. σ σ E [ ε x e G G σ σ / λ x = E [ ε x e σ g σ / λ x ε x e σ g σ / λ x 0 ε ε > 0 and ln x σ λ x g + σ / or ε ε < 0 and ln x σ λ x g + σ / ε g ε εx ln σ λ x εσ. E [ ε x e G G σ σ / λ x =ε x E e g σ and fnally : p = mn [ ε x Φ nf 0 n λ = ε ε σ + g 1 { εg ε εx ln λ x σ ln + εσ Usng the Lagrangan : [ ε εx L = ε x Φ ln σ λ x µ λ + ε ε x λ x ε σ } λ x Φ + εσ ε σ λ x Φ λ ε x Φ ln σ εx ln λ x ε σ εσ εx ln λ x εx λ x εσ εσ

6 6 DUPON OLIVIER the frst-order condtons gve : L λ = x Φ ε σ εx ln λ x εσ mplyng that the arguments of Φ are all equal : for each ε εx ln λ x εσ = d. Consequently ths leadng to : σ λ x = ε x e ε σ σ d µ = 0 ε x e ε σ d σ = ε x e ε σ d σ because σ = 0. As a consequence ε x e ε σ d σ / = 0 = ε x ϕd + ε σ e d. he left-hand term s a decreasng functon of d. Snce not all the ε have the same sgn ts lmts at ± are ± and d s the only soluton of : ε x ϕd + ε σ = 0. Moreover p = mn εx Φd + ε σ λ x Φd 0 n = mn ε x Φd + ε σ ε x Φd 0 n = mn 0 n ε x Φd + ε σ Implementaton. For each d s computed by a bsecton method. he mnmum n then s computed as well as the optmal =. Upper bound approxmaton of the prce therefore s : p = ε x Φd + ε σ Computng the deltas. he same calculus as before shows that : = ε Φd + ε σ x Baset optons. Just as n the case of the lower bound : δ = p S 0 = ε x Φd + ε σ S Asan optons. Lewse : n S0 = =1 p x x. S0

7 APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS 7 References [1 R. Carmona and V. Durrleman. Generalzng the Blac-Scholes formula to multvarate contngent clams. Journal of Computatonal Fnance Volume 9/Number Wnter 005/06. 1